5.1.1 Sampling

5.1.1.1 Instantaneous Sampling

By sampling, an analog signal is converted into a sequence of samples that are usually spaced uniformly in time with a sampling interval of T_s. The choice of the sampling rate (frequency) f_s = 1/T_s, that is, the number of samples per second, is a critical issue. In order that the samples can provide a faithful representation of the sampled analog signal, the sampling rate should increase in proportion with the rate at which the signal varies in time, which determines the signal bandwidth W.

Consider that a baseband signal g(t), band‐limited to 0 < f < W Hz, is sampled at uniform time intervals T_s. Sampling duration is assumed to very short so that the samples may be represented by delta functions located at sampling times nT_s, with −∞ < n < ∞. The sampled signal g_δ(t) may then be expressed as

(5.1)

which follows from (1.21). It is logical that the sampling interval should get shorter as the bandwith occupied by the signal becomes larger, since this implies more rapid time variations in the signal and closer samples are required to reconstruct a faithful replica of the analog signal from its samples.

Since (5.1) may be written as the multiplication of g(t) and an infinite sequence of delta functions, with the following Fourier transform pair (see (1.72)),

(5.2)

the Fourier transform of (5.1), G_δ(f), may be expressed as the convolution of G(f) and H_δ(f) (see (1.46)):

(5.3)

The spectrum of a band‐limited baseband analog signal g(t) sampled at a sampling rate f_s, is thus the superposition of its spectrum frequency shifted at multiples of f_s (see Figure 1.10). Figure 5.3 shows the effect of the sampling rate on the spectrum of the uniformly sampled signal g_δ(t). When the sampling rate is less than 2 W, the signal is said to be under‐sampled. Then, the sampling interval is longer than it should be for recovering the original signal, since the spectra of G(f + nf_s) can not be separated from each other. To solve this problem one needs to shorten the sampling interval and take more samples per unit time, that is, to increase the sampling rate. When the sampling rate is equal to twice the signal bandwidth, f_s = 2 W, then the spectrum G(f + nf_s) does not overlap with neighboring spectra for n − 1 and n + 1, and a low‐pass filter with infinitely sharp cut‐ff may be used to recover the original spectrum G(f) (see Figure 5.4). This sampling rate, f_s = 2 W, that is referred to as Nyquist sampling rate, is considered to be the minimum sampling rate that allows the recovery of the original signal from its samples. The signal is said to be over‐sampled when sampling rate is larger than 2 W. Then, the guard band between the spectra, as shown in Figure 5.3, facilitates the recovery of the original signal without distortion by using a LPF (see Figure 5.4). In practice, the sampling rate is usually taken 10–20% higher than the Shannon’s sampling rate in order to avoid aliasing. For example, a stereo audio system with each of the two channels bandlimited to 20 kHz, is sampled at 44.1 ksamples/s instead of the Nyquist rate 40 ksamples/s.

In summary, an analog signal g(t) band‐limited to W may be recovered from its time samples using a LPF with a passband 0 < f < W, as long as the sampling rate satisfies f_s ≥ 2 W:

(5.4)

Note that G_δ(f) may also be expressed in terms of its time samples at the receiver:

(5.5)

where ℑ[.] denotes the Fourier transform and, consequently, (5.5) is simply the discrete‐time Fourier transform of g_δ(t). Noting the original signal g(t) can be uniquely determined from G_δ(f) (see (5.4)), the time‐samples {g(nT_s)} in (5.5) have all the information contained in g(t) as long as f_s = 1/T_s ≥ 2 W. Hence, a transmitter does not need to transmit an analog signal at all times. Transmission of time‐samples at time intervals T_s ≤ 1/(2 W) is sufficient for exact recovery by the receiver of the original signal g(t) from these samples.

Using (5.4), the original signal g(t) can be reconstructed from the sequence of sample values {g(n/2 W)}:

(5.6)

where sinc(2Wt‐n) is the interpolation function used for reconstructing. Figure 5.5 shows the recovery of an analog signal g(t) from the received quantized samples {1, 3, −1, 1, 3, 1, −3, 1, −1} shown in Figure 5.2b. Interpolation function for each quantized sample is shown in broken line while the recovered analog signal is shown by continuous line. The recovered signal is very similar to the originally transmitted signal shown in Figure 5.2a.

5.1.1.2 Flat‐Top Sampling

In instantaneous sampling, a message signal g(t) is sampled by a periodic train of impulses with sampling period T_s =1/f_s, according to the Shannon’s sampling theorem. The duration of each sample is assumed to be infinitely short so that the samples may be represented by impulses. Instantaneous sampling is not realistic since each sample, represented by an impulse, requires infinite bandwidth. However, in practice, a signal is sampled as a finite duration pulse and the resulting sampled signal has a finite channel bandwidth.

In flat‐top sampling, a sample‐and‐hold circuit (see Example 5.3) is used to sample an analog message signal g(t) to obtain a constant sampled signal level during the sampling time. Hence, the analog signal g(t) is not multiplied by a periodic train of impulses but by a periodic train of rectangular pulses h(t):

(5.7)

Lengthening the duration of each sample to a given value T avoids the use of excessive channel bandwidth, which is inversely proportional to the pulse duration. As shown in Figure 5.2, the duration T of each rectangular pulse h(t) satisfies T/T_s <<1. However, the sampling frequency for flat‐top sampling is still f_s = 1/T_s as in instantaneous sampling.

The expression for a flat‐top sampled signal g_Π(t) may be written as

(5.8)

which implies that the impulses in (5.1) are replaced by h(t)’s. Using (5.3) and the convolution property of the Fourier transform, the Fourier transform of (5.8) is found as

(5.9)

 Noting that the spectrum of the rectangular pulse h(t), which is given by (see (1.16))

(5.10)

is not flat for f < W (see Figure 5.6), low‐pass filtering of a flat‐top sampled signal spectrum (5.9) at the receiver results in a distortion of the original signal spectrum. However, since T <<T_s and the first null of (5.10) is located at f = 1/T, one can write 1/T> > 1/T_s = f_s = 2 W. Therefore, the spectral components of H(f) overlapping with G(f) are practically flat and the distortion caused by H(f) in the recovery of G(f) may be negligibly small. For example, for T/T_s = 0.1 we have WT = 0.05. The magnitude of (5.10) at the edge of the signal bandwidth W is then equal to 0.996 T, which is 0.035 dB below its peak value and hence negligible. Nevertheless, modern communication systems employ an equalizer, in cascade with the low‐pass reconstruction filter, which multiplies the incoming signal with 1/H(f), in order to equalize the first term in (5.9) and to minimize the distortion (see Figure 5.7).

5.1.2 Quantization

Analog signals and their samples vary in a continuous range of amplitudes. Instead of transmitting the amplitudes of the samples as they are, discrete (quantized) amplitude levels may be transmitted. This offers certain advantages in digital communications at the expense of an irrecoverable loss of signal information. However, this loss is controlled by adjusting the spacings between the discrete (quantized) amplitude levels. If the quantization levels are chosen with sufficiently close spacings, the loss will be reduced to tolerable limits and the original signal could be recovered with minimum error.

Amplitude quantization is defined as the process of transforming the sample amplitude g(nT_s) of a message signal g(t) at time t = nT_s into a discrete amplitude v(nT_s) taken from a finite set of possible amplitude levels. The mapping v(nT_s) = q(g(nT_s)) specifies the quantizer characteristics. Quantization process is assumed to be memoryless and instantaneous. This means that the mapping at time t = nT_s is not affected by earlier or later samples of the message signal. Quantizers are symmetric about the origin, that is, for positive and negative values of g(t). All sample amplitude levels g(nT_s) which are located in one of the partition cells are quantized by the corresponding representation level, as shown by Figure 5.10. A quantizer is said to be uniform if the step size is constant and nonuniform if it is not.

5.1.2.1 Uniform Quantization

In a uniform quantizer, decision and representation levels are uniformly spaced, that is, the step size Δ is constant. The number of representation levels L is chosen as a power of 2, that is, L = 2^R, where R is an integer and denotes the number of bits used to represent a quantized sample amplitude. Figure 5.11 shows two different uniform quantizer characteristics, which are of mid‐tread or mid‐rise type.

Let a quantizer maps the input sample g(nT_s), which is a zero‐mean random variable G with continuous amplitude in the range {‐g_max, g_max}, into a discrete random variable V. The peak‐to‐peak signal amplitude 2g_max is divided into L partition cells, each with step size Δ:

(5.15)

Noting that R can assume only positive integer values, L can be 2, 4, 8,... The quantization error is denoted by the random variable Q of sample value q(t) and is characterized by the step size Δ and the number of partition cells L:

(5.16)

G having zero‐mean and the quantizer assumed to be symmetric, V and Q will also have zero‐mean. Assuming that the representation levels are located at the middle of the partition cells, the quantization error Q will be bounded by −Δ/2 ≤ q ≤ Δ/2, since the maximum difference between the input signal sample and its quantized value (representation level) cannot exceed Δ/2 (see Figure 5.10).

If Δ is sufficiently small, then Q may be assumed to a uniformly distributed:

(5.17)

Using the pdf given by (5.17), the variance of the quantization error Q is found to be

(5.18)

Assuming that the thermal noise power is much lower compared to the variance of the quantization error, the output signal to quantization noise ratio, SNR_Q, of a uniform quantizer is defined as the ratio of the average signal power P to the variance of the quantization error:

(5.19)

where α denotes peak to average power ratio (PAPR). Output signal to quantization noise ratio, SNR_Q, is usually expressed in dB:

(5.20)

Note that α = 2 (3 dB) for a sinusoidal signal A cosw_ct since P_peak = A² and P = A²/2. For an audio signal α = 10 dB is usually assumed.

5.1.2.2 Non‐Uniform Quantization and Companding

Some signals have very large PAPRs, for example, PAPR > 10 dB for voice. For example, voice signals, which may be described by Laplacian pdf, assume lower voltage levels more frequently. Therefore, the quantization noise power becomes higher if a uniform quantizer is used for quantizing voice signals. However, if the step size gradually increases with increasing amplitudes, then quantization errors will be higher in infrequent large amplitude ranges but smaller at more frequent amplitudes. This will lead to a drastic decrease in the quantization noise. Therefore, non‐uniform quantization will improve the speech quality. The increased quantization errors associated with large amplitude ranges may still be relatively small when viewed in percentage of the signal’s amplitude. Their impact will not be significant on the perceived speech quality, since ear perceives volume changes logarithmically rather than linearly.

Instead of employing non‐uniform quantization directly, a nonuniform quantizer first compresses an analog signal and then digitizes the compressed signal using uniform quantization. The inverse operation at the receiver consists of converting the quantized signal back into compressed form and then decompressing (expanding) the resulting signal by an expander (see Figure 5.14). Expander used at the receiver restores the signal samples to their correct relative levels, since it performs the inverse of the compression at the receiver. Thus, expander output provides a piecewise linear approximation to the (desired) signal at the compressor input. The combination of compressor and an expander is called a compander.

There are mainly two compression standards for non‐uniform quantization. The so‐called μ‐law compression is commonly used in the United States:

(5.31)

where μ ≥ 0. On the other side, the A‐law compression is widely used in Europe:

(5.32)

where A ≥ 1. Figure 5.15 shows the input‐output characteristics of the μ‐law compressor. As a result of compression, for μ > 0, the SNR for low‐level signals increases at the expense of the SNR for high level signals. A compromise is usually made in choosing the values of μ. Typical values used in practice are μ = 255 and A = 87.6. Note that μ = 0 and A = 1 imply no compression.

5.1.2.3 Signal‐to‐Quantization Noise Ratio with Companding

The variance of the quantization noise may be written as

(5.37)

where a_i, i = 0,1,…,L denote the borders of the decision levels, , and . Noting that the input‐output curve of the nonuniform quantizer has a slope (see Figure 5.21)

(5.38)

where Δ_i and Δ = 2y_max/2^R denote respectively the step sizes before and after companding (see Figure 5.21). Inserting from (5.38) into (5.37), one gets

(5.39)

where the integral expression follows from the previous expression for an infinitely large number quantization levels. The variance of the quantization error for a given compresser characteristics y(x) can be minimized to obtain the optimal compresser [2][3][4][5]:

(5.40)

Inserting the derivative of (5.40) into (5.39), the minimum value of the quantization noise variance is found to be

(5.41)

Using (5.41), the minimum value of the variance of the quantization noise is found to be 0.229 for the Laplacian pdf (5.22) for b=1.

Taking the derivative of y from (5.31) and inserting into (5.39), one can determine the variance of the quantization noise for μ‐law companding as follows:

(5.42)

where Λ = X/x_max has a variance equal to the inverse of the PAPR:

(5.43)

Signal‐to‐quantization noise ratio may then be expressed as

(5.44)

Figure 5.22 shows the variation of the signal‐to‐quantization noise ratio as a function of PAPR for various values of μ. E[Λ] = 0 since the pdf f_X(x) is symmetric around x = 0. For μ = 0, which corresponds to uniform quantization, the SNR_Q decreases linearly with increasing values of the PAPR in log‐log scale. This implies that the system performance does not remain the same over the dynamic range of the signal and rapidly deteriorates with increasing values of PAPR. For example, SNR_Q varies 40 dB for speech signals which typically have 40 dB dynamic range. However, SNR_Q becomes less sensitive to PAPR variations with increasing values of μ and improves at high values of PAPR at the expense of some deterioration at lower PAPR values. For μ = 255, SNR_Q is almost flat for 0 dB < PAPR < 40 dB; hence, equal quality performance is guaranteed for very low and very high volumes of the speech signals.

5.1.3 Encoding

Following sampling and quantization, we obtain discrete‐time and discrete‐amplitude signal samples. The encoding process translates the discrete set of quantized signal samples into a more appropriate form for transmission. The next step in the ADC process is to represent these samples by voltage levels with equal amplitude and opposite polarity, corresponding to binary digits 1 and 0.

5.1.3.1 Differential Encoding

A binary stream may be encoded in an absolute manner or differentially. In absolute encoding, each bit is encoded independently of the others. For example, a binary 1 corresponds to a representation level of +1 V and a binary 0 to −1 V or vice versa. However, differential encoding is not absolute and is based on the transitions in the bit stream to be encoded. For example, consider a bit stream {b_k} to be differentially encoded according to the rule:

(5.45)

where b_k and d_k denote respectively the k^th input and transmitted bits (see Table 5.1). The bar over the binary sum denotes the complement. According to the rule, d_k undergoes a transition (d_k changes from 0 to 1 or from 1 to 0) when b_k = 0, but is unchanged when the input bit b_k = 1. Hence, a transition is used to designate 0 in the incoming binary bit stream {b_k} and a no‐transition is used to designate 1.

k	0	1	2	3	4	5	6	7	8
bk		0	1	1	0	1	0	0	1
dk	1	0	0	0	1	1	0	1	1

Differential encoding requires the use of a reference bit d₀ for initiating the encoding process. Use of 0 or 1 as the reference bit simply inverts the encoded bit stream. Inverting a differentially encoded stream does not affect its interpretation.

Now assume that the differentially encoded bit stream {d_k} in Table 5.1 is received with its third bit in error and an incorrect reference bit is assumed at the receiver. The received bit stream and the incorrect reference bit are shown in bold in Table 5.2. The transmitted bit stream is estimated, in the last line of Table 5.2, by simply observing whether or not a transition has occurred in the polarity of adjacent bits of the received bit stream {d_k}. The use of an incorrect reference bit was observed to alter only the first decoded bit (shown in bold in the last line). On the other hand, the decoded bit stream also shows that a bit received in error causes erroneous decoding of two bits in the decoded bit stream. For differential encoding, each bit error in the received stram {d_k} causes two bit errors in the decoded bit stream {b_k}; hence, the bit error probability is doubled compared to absolute encoding. Table 5.3 shows that the original received bit stream and its inverted version result in the same decoded bit stream. Therefore, differential encoding is immune to the inversion of the polarity of the received bit sequence, that is, to the use of 1 or 0 as the reference bit.

k	0	1	2	3	4	5	6	7	8
dk	0	0	1	0	1	1	0	1	1
bk		1	0	0	0	1	0	0	1

k	0	1	2	3	4	5	6	7	8
dk (original)	1	0	0	0	1	1	0	1	1
dk (inverted)	0	1	1	1	0	0	1	0	0
bk (decoded)		0	1	1	0	1	0	0	1

Despite its degraded bit error performance, differential encoding has the advantage of being immune to polarity inversions. Another advantage of differential encoding is in robust decoding performance. In decoding symbols with absolute encoding, a receiver needs a stable phase reference. However, in some propagation channels, it may not be easy to obtain such a stable phase reference; this may considerably increase the decoding errors. In differential encoding, a receiver does not need a phase reference since transmitted symbols can be estimated by simply comparing the phase of a received symbol with that of the previous one.

5.1.4 Pulse Modulation Schemes

In ADC, sampled and quantized signal samples are encoded by R bits/sample, consisting of 1's and 0's. This is the basis of binary pulse code modulation (PCM), to be explored further in the following sections. In M-ary PCM, k bits are grouped to form a symbol and each symbol is assigned to a different amplitude level. This scheme is called as pulse amplitude modulation (M-ary PAM). For M=2, M-ary PAM reduces to binary PCM. In addition to the PCM, ADC output can be transmitted by three other pulse modulation techniques, namely the pulse amplitude modulation (PAM), the pulse position modulation (PPM) and the pulse duration modulation (PDM) (see Figure 5.23).

In pulse amplitude modulation (PAM), k bits in the encoded bit stream are grouped to form one of M = 2^k amplitude levels. For example, each of M amplitude levels in M‐ary PAM can be transmitted by using pulses with amplitudes ±1 V, ±3 V…. For M = 4, the dibits 00, 01, 11 and 10 may be transmitted by the voltage levels, −3V, −1 V, +1 V, and +3V, respectively. Compared with binary PAM, M‐ary PAM requires higher average transmit power but reduced transmission rate (R = R_b/log₂M), where R and R_b denote, respectively, the transmission rates for M‐ary and binary PAM. For M = 4, the transmission rate and, accordingly, the required channel bandwidth is halved compared to binary PAM. Considering that the transmitted amplitude levels are now ±1 and ±3 compared to ±1 in the binary case, this occurs at the expense of 5‐fold, (1²+3²)/2 = 5, increase in average transmitted power. The receiver estimates the transmitted quantized amplitude level from the received pulse amplitude (see Figure 5.23).

In pulse duration modulation (PDM), the amplitude levels are encoded into the duration of the transmitted pulse. For example, the information of different amplitude levels is transmitted in the channel with pulses of the same amplitude but different pulse durations. The receiver estimates the transmitted voltage level by looking at the duration of the received pulse (see Figure 5.23).

In pulse position modulation (PPM), the amplitude levels are encoded into the position (time of transmission) of transmitted pulses of equal amplitudes and durations. For a 4‐level quantized transmit signal, ±3 V may be transmitted with some shift before/after the reference time of transmission, while ±1 V is transmitted with smaller shift before/after the reference time of transmission. The receiver extracts the information about the amplitude of the transmitted pulse by estimating the time‐shift of the received pulse with respect to the reference transmission time (see Figure 5.23).

5.1.4.1 Binary Versus Gray Encoding

Consider that the bit stream {10010011} is obtained as a result of ADC process. In binary encoding, +1 V is transmitted during T_b (the bit duration) to represent the bit 1 and −1 V for the bit 0, or vice versa. Figure 5.24 shows two different assignments of the above bit sequence where two bits represent a single amplitude level, that is, 4‐ary PAM. Thus, each of the four amplitude levels, ±1 V and ±3 V, is transmitted during 2T_b period for two bits.

In natural encoding shown in Figure 5.24 transmitted amplitude levels are determined by the arithmetic value of the symbol (binary encoding), that is, 00, 01, 10 and 11 correspond to −3 V, −1 V, +1 V, and +3 V, respectively. If the channel noise causes an erroneous reception of the transmitted voltage level, then the error will most likely be between the neighboring voltage levels. In binary encoding, if +1 V is received in error when −1 V is transmitted, the receiver estimates 10 as the transmitted symbol while 01 was actually sent; hence, the receiver suffers two bit errors. Gray encoding assigns voltage levels −3, −1, +1, and +3 volts to 00, 01, 11, and 10, respectively, so that an error in the neighboring received voltage levels causes only a single bit error.

5.1.4.2 Line Codes and the Transmission Bandwidth

Above, we implicitely assumed to use finite‐amplitude pulses (rectangular pulse) to transmit the information carried by binary or M‐ary symbols in the transmission channel. Similarly, binary symbols are assumed to be represented by + V volts and –V volts respectively. However, various other pulses, so‐called line codes, of various shapes, polarity and duration may be used for electrical representation of a bit stream in the transmission channel. Similarly, the bits may be represented in numerous ways other than ± V volts. Figure 5.25 shows several line codes to represent the binary data stream {10010011}, where the bits 0 and 1 are assumed to be equiprobable. Since the PSD of a time‐pulse is proportional to the square of its Fourier transform, the choice of a line code is crucial in determing the transmission bandwidth and intersymbol interference (ISI).

Consider a bit sequence {b_n} encoded by a line code g(t) as follows:

(5.46)

where B = {b_k} is an random binary random stream of equiprobable bits. Noting that (5.46) is in the form of discrete convolution, one may express the PSD of the signal x(t) as the product of the PSD’s of the bit sequence B = {b_k} and g(t) (see Example 1.19):

(5.47)

The autocorrelation function and the PSD of the NRZ polar code are given by (1.147)–(1.49). Here we will first derive the autocorrelation funcion of the NRZ unipolar code, where binary 1 and 0 are equiprobable and are represented by 1 V and 0 V, respectively. The PSD of B = {b_k} will be found by taking the Fourier transform of the autocorrelation function (see Figure 1.11). Since the bits are i.i.d., the autocorrelation of B, which is a power signal, may be written as

(5.48)

The PSD of the power signal B, G_B(f), may easily be found by taking the Fourier transform of R_B[n]:

(5.49)

where the last expression in (5.49) follows from (1.70) or the Poisson sum formula given by (D.78). Assuming that g(t) is a rectangular pulse of amplitude A and duration T_b, then the PSD of g(t) is given by (1.104). Hence inserting (1.104) and (5.49) into (5.47), one gets

(5.50)

The null‐to‐null bandwidth of G_x(t) is 2/T_b since the first null of (5.50) is located at fT_b = ±1. Table 5.4 shows the autocorrelation functions and the PSD’s of some commonly used line codes. The PSDs of the considered line codes are depicted in Figure 5.26. The considered line codes all require infinite transmission bandwidth, even though the PSD of each code is different. For example, Figure 5.26 shows that Manchester code requires a null‐to‐null transmission bandwidth twice as large as the other line codes. One can intuitively justify this since the Manchester code changes twice per bit duration T_b, while the others change only once (see Figure 5.25). In practice, for economic use of the available bandwidth, the spectral components of the line codes with sufficiently low energy content are filtered out, for example, at 3‐dB points. Bearing in mind that the distortion increases as the transmission bandwidth becomes narrower, the transmission bandwidth is usually determined as a result of a tradeoff between the distortion due to filtering and the cost and the availability of the bandwidth.

Line code	RB[n]	GX(f)
NRZ Unipolar
NRZ Polar		A2 Tb sin c2(fTb)
NRZ Bipolar		A2 Tb sin c2(fTb) sin2(πfTb)
Manchester		A2Tb sin c2(fTb/2) sin2(πfTb/2)

5.2.1 Time Division Multiplexing

As shown in Figure 5.27a, the message samples occupy the channel for only a fraction of the sampling interval on a periodic basis. Therefore, the time axis may be used more economically if the time interval between adjacent samples is used by other messages on a time‐shared basis. Thus, time division multiplexing (TDM) enables the joint utilization of a communication channel by several message sources without interference between them. TDM consists of time interleaving of samples from several sources so that the information from these sources can be transmitted serially over a single communication channel (see Figure 5.27b).

A time‐division multiplexer uses a selector switch at the transmitter which sequentially takes samples from each of the signals to be multiplexed and then interleaves these signal samples in the time‐domain as a single high‐speed data stream. At the receiver, this high‐speed data stream is de‐interleaved into low‐speed individual components, by a switch synchronized to the switch at the transmitter, and then delivered to their respective destinations.

Since TDM interleaves N samples, each bandlimited to W, into a time slot equal to one sampling interval T_s ~ 1/2 W (Shannon sampling theorem), it introduces a N‐fold bandwidth expansion, compared to the bandwidth required by a single signal. Consequently, the time duration which may be allocated to each sample must satisfy . The corresponding transmission bandwidth is proportional to , where k is a proportionality constant. Since frequency‐division multiplexing (FDM) of N signals also requires an N‐fold increase in the transmission bandwidth, TDM and FDM have comparable performances as far as their transmission bandwidth requirements are concerned.

Timing operations at the receiver must be synchronized with the corresponding operations at the transmitter. Then, the received multiplexed signals can be properly deinterleaved. Synchronization is therefore essential for satisfactory operation of TDM systems. Its implementation depends on the method of pulse modulation used to transmit multiplexed sequence of samples. The multiplexed signal must include some form of framing so that its individual components can be identified at the receiver. A synchronization word of a given format is usually transmitted within each frame for achieving synchronization.

TDM is highly sensitive to channel dispersion since frequency components of the multiplexed narrow pulses suffer differently from the channel conditions. Therefore, equalization may be required. However, TDM signals are immune to channel nonlinearities because, at a certain time, the channel is accessed only by a single signal sample.

In TDM, analog signals are sampled sequentially at a common sampling rate and then multiplexed over a common channel. By multiplexing digital signals at different bit rates, several digital signals can be combined, for example, computer outputs, digitized voice/facsimile/TV signals, into a single data stream at considerably higher bit rates than any of the inputs.

5.2.2 TDM Hierarchies

Digital multiplexers are commonly used for data transmission services provided by telecommunication operators. A digital hierarchy is used for time‐division multiplexing low‐rate bit streams into much higher‐rate bit streams. By avoiding the use of separate carriers for each signal to be transmitted, TDM provides a cost‐effective utilization of a common communication channel by several independent message sources without mutual interference among them.

Standard PCM representation of a single voice signal, bandlimited to 3.4 kHz, at 64 kbps (8000 samples/s × 8 bits/sample) is called the DS0 Signal. In AT&T’s T1 system, which is the adopted standard in the USA and Canada, first‐level multiplexer combines 24 DS0 streams to obtain a digital signal one (DS1) at 1.544 Mbps, which is higher than 24 × 64 = 1.536 Mbps to account for the overhead and synchronization bits (see Table 5.5). This bit stream is called the primary rate, because it is the lowest bit rate that exists outside a digital switch. The second‐level multiplexer combines four DS1 bit streams to obtain a DS2 at 6.312 Mbps. The third‐level multiplexer combines seven DS2 bit streams to obtain a DS3 at 44.736 Mbps. The fourth‐level multiplexer combines six DS3 bit streams to obtain a DS4 at 274.176 Mbps. The fifth‐level multiplexer combines two DS4 bit streams to obtain a DS5 at 560.160 Mbps. Higher‐level multiplexers are used as the carried traffic increases. The hierarchy of the European E1 system is also shown in Table 5.5.

5.2.3 Statistical Time‐Division Multiplexing

TDM allows multiple users to transmit and receive data simultaneously by assigning the same, fixed number of time slots to each user, even if they have different data rate requirements. TDM works well and is suitable for homogeneous traffic where all users have similar requirements, but does not perform well for different and/or time varying data rate requirements of the users. For example, a busy laser printer shared by many users might need to operate at much higher transmission rates than a personal computer attached to the same line.

Statistical time division multiplexing (STDM) is a method for multiplexing several types of data, with different rates and/or priorities, into a single transmission line. The STDM scheme first analyzes statistics related to the typical workload of each user/device (printer, fax, computer), based on its past and current transmission needs, and then determines in real‐time the number of time slots to allocate to each user/device for transmission. The statistics used in STDM include user/device's peak data rate and the duty factor, that is, the percentage of time the user/device typically spends either transmitting or receiving data. Hence, the STDM uses the total available transmission bandwidth more efficiently than TDM.

5.3.1 PCM Transmitter

In practice, a low‐pass (anti‐aliasing) filter is used at the front‐end of the sampler to filter out signal components at frequencies higher than W, which are assumed to have negligible energy. Following the anti‐aliasing filter, the message signal is sampled with a stream of narrow rectangular pulses (flat‐top sampling). To ensure perfect reconstruction of the message signal at the receiver, the sampling frequency must satisfy f_s > 2 W. Following amplitude quantization, a stream of discrete‐time and discrete‐amplitude samples is obtained. The quantized sample amplitudes are then encoded by assigning R bits to each sample. Following M‐ary PAM modulation, each symbol, consisting of k = log₂M bits, is transmitted using a suitable line code from Table 5.5 at a symbol rate R_s = R_b/k symbols/s (see Figure 5.30).

To alleviate the effects of attenuation, distortion and noise in long ranges, regenerative repeaters may be used in the transmission path. Regenerative repeaters, located at sufficiently close spacing along the transmission path, receive, recover and retransmit the PCM signals. Following a number of repeaters, a PCM receiver, which is also a regenerative repeater, carries out the final regeneration of impaired signals, decoding, regeneration of the stream of quantized samples of the transmitted signal by using a DAC.

Since we already studied the ADC process in detail and a PCM transmitter, which is basically an ADC, we will consider hereafter only the regenerative repeaters and the PCM receiver.

5.3.2 Regenerative Repeater

The repeaters, in analog transmission systems, amplify‐and‐forward not only the input signal but also the input noise (see Figure 5.31). Therefore, effects of noise and distortion from individual links accumulate with increasing ranges. This leads to a gradual decrease of the SNR with the number hops (see Example 4.8). However, regenerative repeaters in PCM receive, and decode the incoming bit stream and retransmit after amplification. Hence, in PCM systems, accumulation of distortion and noise in a cascade of regenerative repeaters is removed (see Figure 5.32). Consequently, except for the delay and some occasional bit errors due to channel noise, the regenerated data stream is exactly the same as the originally transmitted signal. If the repeaters are located sufficiently close to each other, the attenuation and distortion of the received pulses do not reach to a level which cannot be compensated. Repeater spacing can be optimized under the constraints on cost and the received SNR, so that each repeater can successfully detect, remodulate and retransmit the signals at their input. Therefore, PCM systems are used for reliable transmission of digital information over long ranges. Synchronization is of crucial importance for proper operation of PCM systems. Synchronization issues become more serious with increasing data rates due to shorter bit durations and less tolerance to timing errors.

As shown in Figure 5.33, the basic functions performed by a regenerative repeater are equalization, timing and decision making. Equalizer shapes the received pulses so as to compensate for the effects of amplitude and phase distortions. Timing circuitry determines the starting and sampling times of the received pulses as well as their durations so as to generate a clean periodic pulse train at the output. Each sample is compared to a predetermined threshold, at specific time instants (usually in the middle of each bit period), in the decision‐making device so as to decide whether the received symbol is a 0 or 1.

The signal regenerated by the repeater may depart from the original signal for two reasons. Firstly, the channel noise and the interference force the repeater to make incorrect decisions, thus causing bit errors in the regenerated signal. Secondly, if the starting times of the received pulses are strongly affected by noise and/or delay introduced by the channel, a jitter is introduced into the regenerated pulse position, thereby causing distortion.

In a PCM system, the signal can be regenerated as often as necessary. Thus, amplitude, phase and distortions in one hop (between two repeaters) have no effect on the regenerated input signal to the next hop. In that sense, transmission requirements in PCM are independent of range.

5.3.3 PCM Receiver

The receiver, as the last regenerative repeater, amplifies, reshapes and cleans up the effects of noise. The cleaned pulses are regrouped into R‐bit code words and decoded into quantized PAM signal samples. The decoding process consists of regenerating signal samples with amplitudes determined by the weighted sum of the R‐bits in the code word. The weight of each of the R‐bits is determined by its place value (2⁰, 2¹,..., 2^R) in the code word. The message signal is recovered at the receiver by passing the decoder output through a low‐pass reconstruction filter whose cut‐off frequency is equal to the message bandwidth W.

Regenerative repeaters may suffer channel noise and distortion introduced by the quantization process at the transmitter. If the transmission path in one hop is error free, then the recovered signal is not affected by the noise, but the distortion due to quantization process always remains. The quantization noise, introduced at the transmitter, is irrecoverable, since it is signal dependent and disappears only when the message signal is switched off. Quantization noise can be reduced by increasing L = 2^R and selecting a compander matched to the signal characteristics.

5.3.3.1 Receiver Noise

As already studied in Chapter 4, the receiver suffers from internal and external noise, which is always present as long as the system is switched on. Hence, unlike the quantization noise, the receiver noise is independent of the presence of the signal. The noise, which is usually assumed to be AWGN, introduces bit errors depending on the received signal SNR. The bit error performance of a PCM system may hence be improved by using closer repeater spacing, increasing repeater transmit power and/or using low‐noise receivers. Then, the PCM noise performance will be limited by the quantization noise only.

The channel noise may cause bits errors in regenerative repeaters and/or the PCM receiver. Assuming that bits 1 and 0 are transmitted by + A volt and –A volt, respectively, a bit error occurs when the receiver decodes a received bit as 0 when 1 was transmitted or vice versa. When + A volt is transmitted for sending bit 1, the received signal at the receiver may be written as , where w(t) denotes the Gaussian noise voltage at time t, E_b = A²T_b is the bit energy and T_b is the bit duration (This will be studied in detail in Chapter 6). Hence, the pdf of the received signal is Gaussian with mean and variance σ². A bit error occurs when the received voltage level ; then the receiver decodes the received bit as 0. Assuming that the probabilities of transmitting 0 and 1 are equal to each other, that is, p₀ = p₁ = 1/2, the probability of bit error may be written as the area of the tail of the Gaussian pdf from (−∞,0):

(5.51)

where denotes the probability of receiving bit i when bit j is transmitted. From the channel symmetry, . The probability of bit error depends only on the received where E_b denotes the bit energy and N₀ = 2σ² is the one‐sided PSD of the AWGN. The so‐called Gaussian Q function is defined by (see (B.1))

(5.52)

Figure 5.34 shows the variation of the error probability versus E_b/N₀ in dB. Note that Q(0) = 1/2 implies that the bit error probability is upper‐bounded by ½. This corresponds to the case where E_b/N₀ = 0 and the likelihood of receiving 1 or 0 at the receiver is 50% irrespective of the transmitted symbol. Then there is no need for communications and throwing coin is more economical. The bit error probability decreases with increasing values of E_b/N₀ since Q(x) vanishes as x goes to infinity. If the error probability is required to be lower than 10⁻⁸, then the E_b/N₀ should be larger than 12 dB. This corresponds to a single bit error on the average per 10⁸ received bits and also implies that, at a transmission rate of 1 Mbps, one suffers a single bit error in every 100 seconds. To improve the error performance, one evidently needs to increase E_b/N₀, that is, increase the average bit energy, E_b, or decrease the noise PSD, N₀. Even a single bit error per 100 seconds may not be acceptable in some communication systems and techniques other than increasing E_b/N₀ are needed to reduce the bit error probability; these techniques will be studied in the following chapters.

5.3.3.2 SNR at the Decoder Output

The signal‐dependent quantization noise, due to the quantization process at the transmitter, and the receiver noise, which is omnipresent, both contribute to the errors in the decoding process at the receiver. To study the effects of decoding errors, let us assume that a zero‐mean signal with uniform pdf is uniformly quantized, with L quantization levels and step size Δ. The pdf of the allowed signal levels at the decoder input may be written as

(5.55)

which is shown in Figure 5.35.

Using (D.32), the mean‐square signal power at the decoder output is found to be

(5.56)

where denotes the quantization noise power, as given by (5.18). Mean decoded signal to quantization noise power ratio at the decoder output for a uniformly quantized signal with uniform pdf may be written as

(5.57)

The peak decoded signal power level and the corresponding peak decoded signal to quantization noise (power) ratio is given by

(5.58)

which is 3 times (4.78 dB) higher than its mean value.

5.3.3.3 Output SNR of a PCM Receiver

A decoder groups the received bits into R‐bit codewords and reconstructs the quantized samples. These samples are low‐pass filtered and expanded in the case of nonuniform quantization. All bits in a received R‐bit codeword have the same bit error probability P_b which is uniquely determined by the E_b/N₀ as given by (5.51).

In reconstructing the analog waveform of the original message signal, different bit errors are weighted differently since an error in the most significant bit (MSB) in a codeword is more harmful than the one in the least significant bit (LSB). To consider the effects of unequal weighting of the bit errors, we assume that natural binary coding is employed for the binary representation of each of the L quantized values, that is, the lowest quantized level is mapped into a sequence of all zeros; the largest level is mapped into a sequence of all ones; and all other levels are mapped according to their relative values.

Table 5.6 shows in bold some bit errors in the third (least significant) bit, second bit and the first (most significant) bit for a PCM system with L = 8 quantization levels. One may easily observe from Table 5.6 that if an error occurs in the least significant bit, this leads to an error of 2⁰Δ in the representation level. However, if an error occurs in the second bit, its effect on the representation level is 2¹Δ. Finally, an error in the most significant bit causes an error of 2^R‐1Δ in the representation level.

Representation levels	Correctly decoded bits	Error in LSB (3rd bit)	Error in 2nd bit	Error in MSB (1st bit)
+7Δ/2	111	111	111	111
+5Δ/2	110	110	110	110
+3Δ/2	101	101	111	101
+1Δ/2	100	101	100	100
−1Δ/2	011	011	001	011
−3Δ/2	010	010	010	010
−5Δ/2	001	000	001	101
−7Δ/2	000	000	000	100

The possible errors in the decoded representation levels may then be written as

(5.59)

The mean‐square decoding error may then be expressed in terms of the errors in the representation levels as follows:

(5.60)

where the series in (5.60) is evaluated using (D.33). Note that, P_b, and SNR_d are given by (5.51), (5.18) and (5.57), respectively. Since decoding and quantization errors are statistically independent of each other, the two can be added on a power basis to calculate the output SNR:

(5.61)

Note that quantization noise, with variance , is due to quantization of the sample amplitudes at the transmitter, P_b arises from the bit transitions due to the receiver noise and the decoding error, with variance , is due to weighting of the bit errors in the decoding process.

Figure 5.36 shows the variation of SNR_out versus the probability of bit error P_b. Note from Figure 5.34 that P_b = 10⁻², 10⁻⁴, 10⁻⁶ and 10⁻⁸ correspond to E_b/N₀ = 4.3 dB, 8.4 dB, 10.6 dB and 12 dB, respectively. Hence, for small values of the E_b/N₀, P_b is high and the output SNR_out is significantly greater than the E_b/N₀. This is because, at very low input E_b/N₀ values, the noise amplitude is comparable to that of the PCM pulses, and the interpretation of the PCM codeword becomes unreliable. Even a single error in a PCM codeword can change its numerical value by a large amount, the SNR_out in this range decreases rapidly. However, if the E_b/N₀ is sufficiently large, then P_b is low and the total noise is dominated by the quantization process. Then, the output SNR_out, which is proportional to SNR_d ~ 2^2R, improves with R (hence bandwidth) and bandwidth can be exchanged for SNR.

5.4.1 Fundamentals of Differential Quantization

In absolute quantization, each signal sample x[n] is quantized as x_q[n] independently of the other samples and is forwarded to the encoder (see Figure 5.37a). However, in differential quantization, instead of quantizing each sample separately, only the differences between successive time samples are quantized. Since the difference between successive samples is always smaller than the magnitudes of individual samples, the peak‐to‐peak voltage variation and the time rate of change of the difference signal will be smaller. In view of V_pp = LΔ, this implies smaller quantization error (Δ/2) for a given L, or a smaller L for a given value of the quantization error.

The block diagram of a differential quantizer is depicted in Figure 5.37 in comparison with an absolute quantizer. Here, the error signal, which is defined as the difference between a signal sample x[n] and its estimate , is quantized and then forwarded to the encoder. The estimate is obtained by a predictor which estimates the current signal sample using the knowledge of the previous signal samples. The receiver reconstructs the original signal samples using the quantized and encoded error signal.

Figure 5.38 shows samples of a sinusoidal signal, , sampled at , and the error signal where is assumed for simplicity. The peak to peak voltage of x(t) is 2 V while the corresponding value of the error signal is 0.31 V; the average power of the error signal is reduced by 20log(1/0.31) = 10.2 dB compared to x(t). Therefore, for a given value of L, the step size (quantization error) will be smaller for the error signal. Conversely, for a given value of the step size, we can use fewer bits (R), hence narrower transmission bandwidth, to represent the quantized samples. Since the signal x(t) is oversampled, that is, much faster than the Nyquist rate, the resulting encoded signal contains redundant information and adjacent samples are highly correlated.

In summary, differential quantization removes the redundancy due to correlation between adjacent samples. The predictors, which use higher numbers of past samples, may predict the current signal sample more accurately and hence may reduce the peak‐to‐peak variation of the error signal; this allows the use of fewer quantization bits and leads to the reduction of the transmission bandwidth.

5.4.2 Linear Prediction

Depending on the application of interest, the prediction (adaptive filter output) or the prediction‐error may be used as the system output. A prediction filter (predictor) is a finite‐duration impulse response (FIR) filter that provides the best prediction (in some sense) of the present signal sample x[n] of a random signal x(t) using previous samples of the same signal. The prediction represents the desired filter response; previous signal samples supply input to the filter.

Figure 5.39 shows the block diagram of an adaptive FIR filter with a prediction order of p, that is, using p previous time samples of the input signal, x[n‐p],…, x[n‐2], x[n‐1], to provide a linear prediction of the input signal x[n]:

(5.68)

The predicted signal, , is desired to be as close as possible to the input signal x[n]. Hence, the prediction error

(5.69)

is desired to be minimum in some sense. Here, we will determine the effect of the number of taps of the prediction filter in minimizing the prediction error in (5.69) in the mean‐square sense. Ignoring the noise, we first write the variance of the prediction error as

(5.70)

The power of n^th sample x[n] (average input power to the prediction filter) is given by

(5.71)

where the mean value of x[n] is equal to zero. The autocorrelation of x[n] is defined by

(5.72)

The variance of the prediction error may be minimized (the so‐called minimum mean square error, mmse) by differentiating in (5.70) with respect to w_k and equating to zero:

(5.73)

The above is the so‐called Wiener‐Hopf equation for linear prediction. The p unknown filter coefficients w₁, w₂,…, w_p can easily be found from p equations in (5.73). For this purpose, we write the Wiener‐Hopf equation in matrix form as

(5.74)

where

(5.75)

Optimum weight vector which minimizes the mean‐square error is given by

(5.76)

To find the minimum prediction error, we insert (5.73) into (5.70) and get

(5.77)

5.4.3 Differential PCM (DPCM)

DPCM consists of transmitting the prediction error, difference between the actual signal sample and its prediction, as given by (5.69), after quantization (see Figure 5.41). The variance of the error signal is known to be smaller than the variance of the input sample itself (see Figure 5.38 and Example 5.12). Consequently, lower number of quantization levels L, hence fewer bits per sample, may be used to transmit the same signal. Therefore, DPCM allows lower transmission rates and bandwidths.[6]

Figure 5.41 shows the block diagram of a DPCM transceiver. At the transmitter, the signal at the quantizer input is the prediction error, , that is, the difference between the input sample x[n] and its prediction provided by the prediction filter. The prediction error, which is quantized and encoded before transmission, may be reduced by using a predictor with higher prediction order. If prediction is sufficiently good, the variance of e[n] will be less than the variance of x[n], which is simply the input signal power. Note that the simplest prediction filter is a single delay element (a single‐tap filter, p = 1), predicting the present sample as the previous one.

The quantizer output may be written as the sum of the error signal e[n] and the quantization error q[n]:

(5.98)

The quantization error q[n] will be decreased as the number of quantization levels, L, increases. Multilevel quantization of the error signal provides better information for message reconstruction at the receiver. However, with increasing L, the number of bits per sample, , and the required transmission rate, , will increase. Note that the predictor provides a prediction for the n‐th signal sample x[n] by using x_q[n], the quantized version of the n^th signal sample, and the quantized error signal:

(5.99)

It is evident from (5.99) that x_q[n] differs from x[n] only by the irrecoverable quantization error q[n] of the error signal. At the receiver, the decoded quantized error signal is added to the prediction filter output to regenerate the quantized signal sample. The same prediction filter as in the transmitter is used to predict a signal sample from its past values. In the absence of the channel noise, the signal at receiver output x_q [n] differs from the signal sample x[n] only by the quantization error.

A quantizer can be designed to produce a quantization error with variance, , that is smaller than in the standard PCM by choosing a suitable value for the number of quantization levels L. Output SNR of DPCM may be expressed in terms of the variance of the prediction error as

(5.100)

where denotes the variance of the input sample x[n], is the variance of the quantization error q[n] and the variance of the prediction error. According to (5.19), the signal‐to‐quantization noise ratio is given by . The processing gain G_p due to differential quantization

(5.101)

provides an SNR advantage to DPCM over standard PCM in the order of 5–10 dB for voice signals. [6]

The error is proportional to the derivative of the input signal. Since changes as much as ± (L‐1)Δ from sample to sample (see Table 5.6), the maximum rate of increase of the error signal should be at least as fast as the input sequence of samples {x[n]} in a region of maximum slope. Hence, the DPCM signal should satisfy the following slope tracking condition:

(5.102)

where f_s = 1/T_s is the sampling frequency and Δ is the step size. Otherwise, Δ may be too small for the error signal to follow a steep segment of the input signal x(t); then we suffer the so‐called slope overload distortion. On the other hand, when the step size Δ is too large relative to local slope of input signal x(t), granular noise causes oscillations around relatively flat segments of signal samples. Since small values of the step size causes slope overload distortion, while large values of it is the source of granular noise, one should seek for an optimum step size which provides a trade‐off between the slope overload distortion and the granular noise.

5.4.4 Delta Modulation

Delta modulation (DM) is basically a special case of DPCM with three fundamental differences. In the block diagram of DPCM shown in Figure 5.41:

1. The message signal is oversampled (higher than Nyquist rate) to purposely increase the correlation between adjacent samples. The rationale behind oversampling is to ensure that adjacent samples do not differ from each other significantly.
2. The predictor has a prediction order p = 1 (single‐tap) in DM but p ≥ 2 for DPCM (multi‐tap). Since we have at the output of the single‐tap predictor, the error signal defined by (5.69) may be rewritten as
   (5.103)
    The quantized sample x_q[n] in DM differs from the original sample x[n] by the quantization error q[n] as defined by (5.99).
3. The error signal e[n] in (5.103) is quantized with one‐bit; the quantization level is L = 2 for DM compared to multilevel (L > 2) quantization in DPCM. As also shown in Figure 5.42, depending on positive (representing bit 1) and negative signs (representing bit 0) of e[n], the quantized error signal assumes one of two levels, ±Δ:
   (5.104)

DM provides a staircase approximation to the oversampled version of the message signal and achieves digital transmission of analog signals using a much simpler hardware than PCM. Since each sample is quantized with one bit, transmission rate in DM is equal to the sampling rate f_s = 1/T_s, which is chosen to be much higher than the Nyquist rate, f_s = 2 W. Table 5.7 shows the process of DM of the following sample sequence: x[1] = 0.9, x[3] = 0.3, x[4] = 1.2, x[5] = −0.2, x[6] = −0.8.

n	x[n]			eq[n]
1	0.9	0 (ref.)	+0.9	+1	1
2	0.3	1	−0.7	−1	0
3	1.2	0	+1.2	+1	1
4	−0.2	1	−1.2	−1	0
5	−0.8	0	−0.8	−1	−1

In a DM receiver, the quantized error signal is added to the one‐tap predictor output successively so that the n‐th sample of the quantized signal is found as follows:

(5.105)

Thus, at the n‐th sampling instant, the accumulator increments the predictor output by ∆ in positive or negative direction depending on the sign of e_q[n]. The staircase approximation x_q(t) is reconstructed by passing the sequence of positive/negative pulses, produced at the decoder output, through an accumulator in a manner similar to that used at the transmitter. Note that (5.105) is the digital equivalent of integration, since it represents the accumulation of positive and negative increments of the step size ∆. Out‐of‐band quantization noise in x_q(t) is rejected by passing it through a LPF, of a bandwidth equal to the message bandwidth.

5.4.4.1 Slope‐Overload Distortion

In view of (5.99) and (5.103), the error signal at the quantizer input may also be written as

(5.106)

Ignoring the quantization error q[n‐1], the error signal is proportional to the derivative of the input signal. Since the maximum rate of increase of the error signal is Δ/T_s and occurs with a sequence of Δ’s of the same polarity (see Figure 5.43), the sequence {x_q[n]} can increase faster than the input sequence of samples {x[n]} in a region of maximum slope if

(5.107)

Otherwise, the slope Δ/T_s of x_q(t) cannot follow a steep segment of the input signal x(t); this leads to slope overload distortion as shown in Figure 5.43.

5.4.4.3 Adaptive Delta Modulation (ADM)

ADM comprises an additional hardware, compared to DM, designed to provide variable step size, thereby reducing slope overload effects without increasing the granular noise. Slope overload appears in e_q[n] as a sequence of pulses having the same polarity, whereas granular noise occurs when the polarity tends to alternate when x_q(t) tracks x(t).

As shown in Figure 5.44a, which shows the block diagram of an ADM, the step‐size controller adjusts the variable gain g(n), so that (5.105) is modified as

(5.110)

where the step size is adaptively determined as follows:

(5.111)

Here, K is a constant in the range 1 < K < 2. In view of (5.110) and (5.111), the effective step size increases in discrete steps by successive powers of K during slope overload conditions. Variable step size yields a wider dynamic range. The SNR of ADM is typically 8–14 dB better than ordinary DM and the transmission bandwidth for voice is typically 24–32 kHz [6]. A scheme known as continuously variable slope delta modulation (CVSD) provides a continuous range of step‐size adjustments instead of a set of discrete values as in (5.111). Consequently, CVSD has better performance than DM and ADM.

5.4.4.4 Delta‐Sigma Modulation (DΣM)

As can be observed from (5.106), the error signal at the quantizer input of a DM is proportional to the derivative of the message signal. This leads to accumulation of noise power in the demodulated signal due to the presence of accumulator (integrator) in the DM receiver (see Figure 5.41). This drawback can be overcome by integrating the message signal prior to DM at the transmitter as shown in Figure 5.45. By integration at the transmitter, the low‐frequency content of the input signal is pre‐emphasized and the correlation between adjacent samples of the DM input is increased. The overall system performance is thus improved by reducing the variance of the error signal at the quantizer input and, at the same time, the receiver design is simplified.

The so‐called delta‐sigma modulation (DΣM) is a version of DM that incorporates integration at its input. Since integration at the transmitter input should be compensated for by differentiation at the receiver, this need is satisfied by simply removing the integrator in the conventional DM receiver. Thus, as shown in Figure 5.45, the receiver of a DΣM consists only of a LPF.

Simplicity of implementation of the DΣM transceiver is achieved at the expense of sampling rates far in excess of that needed for PCM, hence increased transmission bandwidth. Then, one may use, for example, a multi‐tap prediction filter as in DPCM in order to reduce the transmission bandwidth.

5.4.5 Audio Coding

A voice signal, bandlimited to 3.4 kHz, sampled at 8 ksamples/s and quantizated with 8‐bits, is transmitted at 64 kbps. This rate is unacceptably high for some applications that require much lower data rates.

The speech process starts with the generation of a tone in the vocal chords. The throat, mouth, tongue, lips, and nose, and so on, acting as filters, modulate the generated tone. The speech signal consists of active periods separated by unvoiced time intervals. Unvoiced speech is modelled by white noise. Voiced speech may be modelled by periodic sequence of impulses with period which is equal to the reciprocal of the pitch frequency. Figure 5.46 shows a model for the generation of speech signals.

N samples of a speech signal are predicted as

(5.112)

where w_n denotes the input sequence (white noise or impulses), G is the amplifier gain, {a_i} are the tap gains and p denotes the prediction order. The first part in (5.112) represents the output of a prediction filter with p taps as shown in Figure 5.37. However the term Gw_n does not depend on the previous speech samples. Linear prediction is used to determine the tap gains {a_i}, as outlined in Section 5.4.2. For the optimal choice of the predictor coefficients, the mean square value of the prediction error may be written as

(5.113)

When the excitation coefficients are normalized to unity, then the gain G is given by [2]

(5.114)

Linear predictive coding (LPC) provides an alternative approach to digital transmission of analog voice signals compared to the classical approach, which is based on ADC of the analog voice signals. LPC is based on extracting the parameters of a voice signal, for every 10‐25 ms, and transmitting this information with a prespecified number of bits. A transversal filter and some auxiliary components are used to extract the required signal parameters, for example, volume and the duration of the tone, pitch frequency, and so on, and the tap weights of the transversal filter. [2][6] As shown in Figure 5.47, voice samples are analyzed to determine the parameters for the synthesizer. The error signal, defined as the difference between the input and synthesizer output, is encoded along with these parameters for transmission. These parameters and the quantized error signal are used by the receiver to regenerate the original speech signal.

A complete LPC codeword consists of typically 80 bits accounting for pitch frequency, amplifier gain, tap weights of the transversal filter, and so on. Updating the parameters every 10‐25 ms is equivalent to sampling at 40‐100 Hz. So, LPC requires very modest bit rates in the range of 3.2 kbps (80 bits/25 ms)‐8 kbps (80 bits/ 10 ms). [6] Therefore, LPC is very effective for digital voice transmission and is currently employed in many modern communication systems.

The GSM system uses regular pulse excited long‐term prediction (RPE‐LTP) coding, which is based on PLC. ADC of speech signals at 8 ksamples/s and 13 bits/sample leads to a bit rate 104 kbps at the RTP‐LTP encoder input. Based on the observation that the speech signals are approximately constant during 20 ms intervals, the filter parameters characterizing a speech signal are updated with 20 ms intervals. This corresponds to sampling speech signal at 1/20 ms = 50 samples/s; such low sampling rates provide a compromise between the speech quality and the bit rate. In the RPE‐LPT algorithm, features of each 20 ms speech block are extracted, and encoded into 260 bits with a net bit rate of 260 bits/20 ms = 13 kbits/s. Hence, a RTP‐LTP encoder reduces the input bit rate of 104 kbps to 13 kbps. Speech signals are then protected against channel errors with the addition of 196 redundancy bits, so that a total of 456 bits are transmitted during 20 ms interval with a channel rate of 456 bits/20 ms = 22.8 kbits/s. Note that this rate is approximately ~22% of 104 kbps obtained using ADC.

The adaptive multi‐rate (AMR) is a multi‐rate narrowband audio compression format optimized for speech coding. It is based on 160 bits for 20 ms frames, and narrowband (200–3400 Hz) signals are encoded at variable bit rates ranging from 4.75 to 12.2 kbit/s. AMR was adopted as the standard speech codec by 3GPP in October 1999 and is now widely used in GSM and UMTS. It adaptatively selects one of eight different bit rates (12.2, 10.2, 7.95, 7.40, 6.70, 5.90, 5.15 or 4.75 kbit/s) depending on the channel conditions. [7][8] Although AMR is basically a speech format and may not give ideal results for other audio signals, AMR codec is also used for storing short audio recordings in mobile telephone handsets. The AMR codec can also be used for applications like random access or synchronization with video.

5.4.6 Video Coding

Multimedia communications in many applications require transmission of audio, video and data signals at low data rates. We know from Example 5.5 that high data rates are needed to transmit digital video services unless video compression is used. As an integral part of video encoding, high compression ratios of video signals are strongly desired in various applications including TV broadcasting, videoconferencing, digital storage, internet streaming, and wireless communications.

Data compression drastically reduces the transmission rate and enables efficient storage and transmission of multimedia signals. Lossless data compression (data compaction) is reversible and operates as ZIP codes used to compress document files. The objective of lossless data compression is to map the original bit stream into a shorter stream, so that the original bit stream can be recovered exactly from the compressed stream. Lossy data compression is irreversible, and aims to compress the original data at the expense of some degradation.

Video compression is possible due to the fact that adjacent parts of a single video frame as well as successive frames are similar to each other. Therefore, transmitting the differences between scenes within the frame and between adjacent frames is much more economical than transmission of all the frames. The interframe redundancy can be significantly reduced to produce a more efficiently compressed video signal. This reduction is achieved through the use of prediction of each frame from its neighbors; the resulting prediction error (differential encoding) is transmitted for motion estimation and compensation. In brief, video compression algorithms exploit two types of redundancy in the signal: temporal (changes from one frame to the next) and spatial (changes in the pixels within one frame). Still image coding algorithms like Joint Photographic Experts Group (JPEG) only exploit the spatial redundancy.

Video compression techniques include

1. Discrete cosine transform (DCT), which is a frequency transform technique similar to Fourier Transform, is used to reduce the spatial redundancy in video frames.
2. Quantization is a technique for deliberately losing some information (lossy compression) from the pictures.
3. Huffman coding, run length coding, and variable length coding are lossless compression techniques and use code tables based on the data statistics to encode more frequently encountered data with shorter codewords.
4. Motion compensated predictive coding aims to exploit temporal redundancy. The differences (changes) between current and preceeding frames are calculated and encoded.
5. Bi‐directional prediction where some images are predicted from the pictures immediately preceding and following the image.

The first three of the above compression techniques are also used by the JPEG standard, which was developed for compressing full‐color or greyscale images of natural, real‐world scenes by exploiting known limitations of the human visual system. Low resolution pictures such as computer generated images can be transmitted or stored at data rates up to 1.5 Mbps.

Motion Photographic Experts Group (MPEG‐1) standard was initially designed to primarily compress video‐CD (non‐interlaced video) images of size 352 × 288 at 25 frames/s at a transmission rate of 1.5 Mbps. In contrast with JPEG, MPEG‐1 exploits the interframe (temporal) redundancy and provides motion‐compensated compression. [9]

MPEG‐2 standard, published in 1995, was designed for coding standard‐definition (SD) 4:2:0 video at bit rates 3‐6 Mbit/s, with a drastic reduction from ~144 Mbps. MPEG‐2, used for digital TV broadcast and DVD, has many features in common with MPEG‐1 and achieve compressing ratios up to 50:1.

MPEG‐3, launched mainly for the HDTV applications, refers to a group of audio and video coding standards designed to handle HDTV signals at 1080p (p stands for progressive scanning) in the range of 20 to 40 Mbps. However, as MPEG‐2 was observed to have the ability to accommodate HDTV services as well, HDTV was included in 1992 as a separate profile in the MPEG‐2 standard.

MPEG‐4, though initially introduced in 1998 for low bit‐rate video communications, is efficient across a variety of bit‐rates ranging from a few kbps to tens of Mbps. MPEG‐4 provides improved coding efficiency over MPEG‐2, ability to encode multimedia data (video, audio, speech), and robustness against errors. For MPEG‐4, bit rates as low as 2 Mbp/s is considered to provide a good compromise between picture quality and transmission bandwidth efficiency. MPEG‐4 addresses speech and video synthesis, fractal geometry, computer visualisation, and an artificial intelligence approach for reconstructing images.

ITU‐T H.264/MPEG‐4 Advanced video coding (AVC) standard, which is published in 2003, allows broadcasting over cable, satellite, cable modem, terrestrial, and so on. Using H.264/AVC, it is possible to transmit video signals at about 1 Mbps with PAL quality, which enables streaming over xDSL connections. H.264 standard contains a set of video coding tools but not all of them are needed for every application. A H.264 decoder may choose to implement any number of them depending on the needs of the application. The adoption of H.264 standard led to doubling the number of TV programmes per satellite in comparison to the use of H.262 (MPEG‐2). H.262 also allows

1. Interactive or serial storage multimedia on DVD, optical and magnetic devices,
2. Videoconferencing and videophone,
3. Video‐on‐demand and multimedia streaming services,
4. Multimedia messaging services, and
5. Multimedia services over packet services, such as multimedia mailing.

High efficiency video coding (HEVC) standard was published in 2013. It aims to achieve a factor of two improvement in compression efficiency compared to the H.264/AVC standard. HEVC is a strong candidate for new generation digital TV services using 1080p.

Figure 5.48 shows the milestones in the evolution of the bit rate required for the transmission of HD(1080p) TV signals. One may easily observe drastic decrease of the required bit rates in this rapidly evolving area.