A channel equalizer whose parameters are updated automatically and adaptively during transmission of data.

The channel whose model is that of corrupting a transmitted waveform by the addition of white (i.e., spectrally flat) Gaussian noise.

The complex inner product of the sequence with a shifted version itself.

The spectrum of the waveform is nonzero for frequencies in some band concentrated about a frequency f_c ≥ 0; f_c is called the carrier frequency.

Transmission efficiency of a digital modulation scheme measured in units of bits per second per Hertz of bandwidth.

A signal with frequency content centered around DC.

The spectrum of the waveform is nonzero for frequencies near f = 0.

An independent set of vectors in a linear space in terms of which any vector in the space can be represented as a linear combination.

The statement that the norm squared of any vector in a vector space is greater or equal to the sum of the squares of the projections onto a set of orthonormal basis vectors.

A mapping from a set of messages into binary strings.

The period of time over which a single symbol is transmitted.

A convergent sequence whose members become progressively closer as the limit is approached.

The highest rate at which information can be transmitted reliably across a channel.

A device that is used to reduce the effects of channel distortion in a received signal.

The medium over which communication signals are transmitted. Examples are fiber optic cables, free space, or telephone lines.

The idea that a basis set is extensive enough to represent any vector in the space as a Fourier sum.

The function g(t) of a bandpass waveform v(t) where the bandpass waveform is described by

Frequency modulation where the phase varies in a continuous manner.

A sequence of elements {x_k} in a metric space converges to a limit x if $\underset{k\to \infty }{\lim }d(x,x_k)=0$ where d(x,x_k) is the metric in the space.

The optimal receiver structure for digital communications in AWGN.

A set that can be put in one-to-one correspondence with the positive integers.

The complex inner product of the first sequence with a shifted version of the second sequence.

The boundary in signal space between the various regions where the receiver declares H_i. Typically, a hyperplane when dealing with AWGN channels.

Mode for adjustment of the equalizer coefficient adaptively based on the use of the detected symbols at the output of the detector.

An adaptive equalizer that consists of a feedforward filter and a feedback filter, where the latter is fed with previously detected symbols that are used to eliminate the intersymbol interference due to the tail in the channel impulse response.

A vector space is n-dimensional if it possesses a set of n independent vectors, but every set of n + 1 vectors is linearly dependent.

A channel model characterized by discrete input and output alphabets and a probability mass function on the output conditioned on the input.

A channel that elongates and distorts the transmitted signal. Normally modeled as a time-varying linear system.

A measure of the average uncertainty of a random variable. For a random variable with distribution p(x), the entropy H(X) is defined as −∑_xp(x)logp(x).

A discrete-time transfer function (z transform) that relates the transmitted amplitudes to received samples in the absence of noise.

That percentage of the baseband transmitted spectrum which is not contained within the Nyquist band.

Superposition of segments of a received PAM signal that indicates the amount of intersymbol interference present.

An approximation of a function in a Hilbert space as a linear combination of orthonormal basis functions where the coefficient of each basis function is the projection of the function onto the respective basis function.

If w(t) is a waveform, then the Fourier transform of w(t) is

where f has units of hertz.

A tapped-delay line channel equalizer in which the delay between adjacent taps is less than the duration of a transmitted symbol.

A digital modulation technique in which the transmitted pulse is sinusoidal, where the frequency is determined by the source bits.

MSK where the data signal is prefiltered with a Gaussian filter prior to FM.

An algorithm that produces a set of orthonormal vectors from a linearly independent set of vectors.

A normed linear space complete in its natural metric.

A procedure that constructs a code of minimum average length for a random variable.

An element in a vector space that when multiplied by any vector reproduces that vector.

A set of vectors is independent if any one of them cannot be expressed as a linear combination of the others.

A function of ordered pairs of vectors in a vector space, which is analogous to the dot product in ordinary Euclidean vector space.

The ill-effect of one symbol smearing into adjacent symbols thus interfering with the detection process. This is a consequence of the channel filtering the transmitted signals and therefore elongating their duration, see dispersive channel.

A representation for second-order random processes that allows one to express a random process in terms of a superposition of deterministic waveforms. The scale values are uncorrelated random variables obtained from the waveform.

A procedure for coding that does not use the probability distribution of the source but nevertheless is asymptotically optimal.

Linear sum of a set of vectors in a vector space with each member in the sum, in general, multiplied by a different constant.

A subspace of a Hilbert space that contains the origin and is closed under addition of elements.

A set of vectors that is not linearly independent.

A set of vectors, none of which can be expressed as a linear combination of the others.

See stochastic gradient algorithm.

The receiver filter with impulse response equal to the time-reversed, complex conjugate impulse response of the combined transmitter filter-channel impulse response.

An estimate for a parameter that maximizes the likelihood function.

A detector for estimating the most probable sequence of data symbols by maximizing the likelihood function of the received signal.

A periodic binary {0,1} sequence that is generated by a shift register with linear feedback and which has maximal possible period given the number of stages in the shift register.

Two random vectors or time-limited waveforms are mean-square equivalent if and only if the expected value of their mean-square error is zero.

Estimation of a random variable by a linear combination of n other random variables that minimizes the expectation of the squared difference between the random variable to be estimated and the approximating linear combination.

A real-valued function of two vectors in a vector space that is analogous to the distance between them in the ordinary Euclidean sense.

A vector space in which a metric is defined.

A special form of CPM having linear phase trajectories and a modulation index of 1/2.

The bandpass signal

where fluctuations of g(t) are caused by the information source such as audio, video, or data.

The information source, m(t), that causes fluctuations in a bandpass signal.

A measure for the amount of information that a random variable gives about another.

A function of a single vector in a vector space that is analogous to the length of a vector in ordinary Euclidean vector space.

A vector space in which a norm has been defined.

The narrowest frequency band that can support a PAM signal without intersymbol interference (the interval [–1/(2T), 1/(2T)] where 1/T is the symbol rate).

A condition on the overall frequency response of a PAM system that ensures the absence of intersymbol interference.

The space of all possible received data vectors in a signal detection or estimation problem.

The property of two vectors expressed by their inner product being zero.

The space of vectors that are orthogonal to a linear manifold or subspace.

Multiple access technique in which a different subset of OFDM subcarriers are assigned to individual users, leading to a multicarrier transmission scheme.

Multicarrier modulation by using a collection of low-bit-rate orthogonal subcarriers.

A theorem stating that the minimum mean-square estimate of a random variable in terms of a linear combination of n other random variables requires the difference between the random variable and linear combination, or error, to be statistically orthogonal to each random variable in the linear combination (i.e., the expectation of the product of the error and each random variable is zero), also called the projection theorem.

The property of two or more vectors or time-limited waveforms being mutually orthogonal and individually having unit length.

A basis set for which the basis vectors are orthonormal.

The probability of failing to achieve adequate reception of the signal due to, for instance, co-channel interference.

The statement that the norm squared of a vector in a complete Hilbert space equals the sum of the squares of the vector's projections onto an orthonormal basis set in the space.

A signaling technique in which a controlled amount of intersymbol interference is introduced at the transmitter in order to shape the transmitted spectrum.

Modulation where the instantaneous phase of the carrier varies linearly with the data signal.

Received SINR that is required to achieve reliable communication with specified bandwidth efficiency.

Relative power in a modulated signal as a function of frequency.

A transformation of source symbols at the transmitter that compensates for intersymbol interference introduced by the channel.

A channel equalizer whose parameters are fixed (time-invariant) during transmission of data.

See orthogonality principle.

Also referred to as pseudorandom sequences (PN), these are sequences that are deterministically generated and yet possess some properties that one would expect to find in randomly generated sequences.

A digital modulation technique in which the source bits are mapped to a sequence of amplitudes that modulate a transmitted pulse.

A serial bit stream that consists of binary words which represent quantized sample values of an analog signal.

Modulation where information is transmitted in the amplitude of the cosine and sine components of the carrier.

Replacing a sample value with the closest allowed value.

A pulse shape with Fourier transform that decays to zero according to a raised cosine; see Equation 3.18. The amount of excess bandwidth is conveniently determined by a single parameter (α).

The minimum rate at which a source can be described to the given average distortion.

A channel that randomly scales the transmitted waveform by a Rayleigh random variable while adding an independent uniform phase to the carrier.

The function R(t) = |g(t)| of a bandpass waveform v(t) where the bandpass waveform is described by

A theorem stating the conditions under which an element of a space of square-integrable functions can be represented in terms of an infinite orthonormal set.

A pulse shape with Fourier transform that is the square root of the raised cosine spectrum shown in Equation 3.18.

An inequality expressing the fact that the absolute value of the inner product of any pair of vectors in a Hilbert space is less than or equal to the product of their respective norms.

A Hilbert space in which a countable set of elements exists that can be used to represent any element in the space to any degree of accuracy desired as a linear combination of the members of the set.

A sequence with symbols drawn from a field, which satisfies a linear-recurrence relation and that can be implemented using a shift register.

The permitted values of the complex envelope for a digital modulating source.

An abstraction for representing a time-limited waveform in a low-dimensional vector space. Usually arrived at through the application of the Karhunen–Loève transformation.

A vector representing a received signal in a signal detection or estimation problem.

Multiple access technique in which non-overlapping subcarriers are assigned to different users, leading to a single carrier transmit signal.

The set of elements of a separable Hilbert space used to represent an arbitrary element with any degree of accuracy desired.

A measure of how efficiently space, frequency, and time are used. It is expressed in erlang per square meter per hertz.

A signaling technique in which the pulse bandwidth is many times wider than the Nyquist bandwidth.

An algorithm for adaptively adjusting the coefficients of an equalizer based on the use of (noise-corrupted) estimates of the gradients.

A tapped-delay line channel equalizer in which the delay between adjacent taps is equal to the duration of a transmitted symbol.

The probability of classifying the received waveform into any of the symbols that were not transmitted over a particular bit interval.

Mode for adjustment of the equalizer coefficients based on the transmission of a known sequence of transmitted symbols.

An inequality of a normed linear space that is analogous to the fact that the length of the vector sum of any two sides of a triangle is less than or equal to the sum of their respective lengths.

A function relating the number of subscribers per channel and the number of channels per cell for different values of blocking probability.

An element of a linear, or vector, space.

A space of elements, called vectors, which obey certain laws of associativity and commutativity and have identity elements for scalar multiplication and element addition.

A filter whose output at a given time depends not only linearly of present and past inputs, but on powers and products of present and past inputs.

See wavelet transform.

Resolution of a signal into a set of basis functions called wavelets. As a parameter of the wavelet is changed, the behavior of the signal over progressively shorter time intervals is resolved.

A design constraint which specifies that intersymbol interference be eliminated.

A channel equalizer whose parameters are adjusted to completely eliminate intersymbol interference in a sequence of transmitted data symbols.