Time Domain Equalization and Interference Cancelation

Abstract

This chapter concentrates on the modeling of intersymbol interference (ISI) channels and the various signal processing methods for recovering digital information transmitted over such channels. The chapter begins with a treatment of ISI channel modeling, including a vector representation of digital signaling on ISI channels. The maximum likelihood receiver is then developed for ISI channels, leading to an equivalent model of the ISI channel known as the discrete-time white noise channel model. The effects of using fractional sampling or over-sampling at the receiver are also considered, where the sampling rate is an integer multiple of the modulated symbol rate. Afterwards, the various time domain equalization techniques are considered, including the linear zero-forcing and minimum mean-square-error equalizers, and the nonlinear decision feedback equalizer. Afterwards, sequence estimators are considered beginning with maximum likelihood sequence estimation (MLSE) and the Viterbi algorithm. Since the MLSE receiver can have high complexity for channels that have a long impulse response, some reduced complexity sequence estimation techniques are considered such as reduced state sequence estimation (RSSE) and delayed decision feedback sequence estimation (DDFSE). The chapter goes on to provide an analysis of the bit error rate performance of MLSE on static ISI channels and multipath-fading ISI channels, and fractionally-spaced MLSE receivers for ISI channels. Finally, the chapter provides a discussion of co-channel demodulation for digital signals on ISI channels, and concludes with a receiver structure that incorporates a combination of optimal combining and sequence estimation as implemented with the Viterbi algorithm.

Appendices

Appendix 7A: Derivation of Eq. (7.202)

Consider the case where D = 1 and suppose that the characteristic function in (7.194) has L + 1 different poles $\bar{\boldsymbol{\alpha }}= (\bar{\alpha }_{0},\bar{\alpha }_{1},\ldots,\bar{\alpha }_{L})$. Then the pairwise error probability is equal to

$$\displaystyle{ P(\bar{\boldsymbol{\alpha }}) =\sum _{ i=0}^{L}\left (\left (\frac{1} {2} -\frac{1} {2}\sqrt{ \frac{\bar{\alpha }_{i } } {1 +\bar{\alpha } _{i}}}\right )\prod _{j\neq i}\left (1 -\frac{\bar{\alpha }_{j}} {\bar{\alpha }_{i}} \right )^{-1}\right ). }$$

(7A.1)

Define the function $\phi (\bar{\boldsymbol{\alpha }}) =\sum _{ i=0}^{L}\bar{\alpha }_{i} - C = 0$, where C is a constant. The method of Lagrange multipliers suggests that

$$\displaystyle{ \frac{\partial P(\bar{\boldsymbol{\alpha }})} {\partial \bar{\alpha }_{i}} +\lambda \frac{\partial \phi } {\partial \bar{\alpha }_{i}}\ = 0\ \ i\ = 1,\ \ldots,\ L }$$

(7A.2)

for any real number λ. It can be shown by induction that

$$\displaystyle\begin{array}{rcl} \frac{\partial P(\bar{\boldsymbol{\alpha }})} {\partial \bar{\alpha }_{k}} & =& -\left (\frac{1} {2} -\frac{1} {2}\sqrt{ \frac{\bar{\alpha }_{k } } {1 +\bar{\alpha } _{k}}}\right )\sum _{i\neq k}\left ( \frac{\bar{\alpha }_{i}} {\bar{\alpha }_{k}^{2}}\left (1 - \frac{\bar{\alpha }_{i}} {\bar{\alpha }_{k}}\right )^{-2}\prod _{ j\neq i,k}\left (1 -\frac{\bar{\alpha }_{j}} {\bar{\alpha }_{k}}\right )^{-1}\right ) \\ & \ & +\sum _{i\neq k}\left (\frac{1} {\bar{\alpha }_{i}} \left (1 -\frac{\bar{\alpha }_{k}} {\bar{\alpha }_{i}} \right )^{-2}\left (\frac{1} {2} -\frac{1} {2}\sqrt{ \frac{\bar{\alpha }_{i } } {1 +\bar{\alpha } _{i}}}\right )\prod _{j\neq i,k}\left (1 -\frac{\bar{\alpha }_{j}} {\bar{\alpha }_{i}} \right )^{-1}\right ) \\ & \ & -\left ( \frac{1} {4\bar{\alpha }_{k}^{1/2}} \frac{1} {(1 +\bar{\alpha } _{k})^{3/2}}\right )\prod _{j\neq k}\left (1 -\frac{\bar{\alpha }_{j}} {\bar{\alpha }_{k}}\right )^{-1}. {}\end{array}$$

(7A.3)

By solving (7A.2) and observing the symmetry of $P(\bar{\boldsymbol{\alpha }})$ and the derivative (7A.3) with respect to the permutations of $\bar{\boldsymbol{\alpha }}$, it is apparent that the minimum of $P(\bar{\boldsymbol{\alpha }})$ is achieved when $\bar{\alpha }_{0} =\bar{\alpha } _{1} =\ldots =\bar{\alpha } _{L}$.

Problems

7.1. Starting with

$$\displaystyle{f_{k} =\int _{ -\infty }^{\infty }h^{{\ast}}(\tau )h(\tau +kT)d\tau }$$

show that

$$\displaystyle{F(\mathrm{e}^{j2\pi fT}) = F_{\varSigma }(f).}$$

7.2. Suppose that the impulse response of an overall channel consisting of the transmit filter, channel, and receive filter is

$$\displaystyle{F(f) = \left \{\begin{array}{ll} 1\ &,\ \ \vert f\vert \leq f_{\ell} \\ \frac{f_{u}-\vert f\vert } {f_{u}-f_{\ell}} \ &,\ \ f_{\ell} \leq \vert f\vert \leq f_{u} \end{array} \right..}$$

• (a) Find the overall impulse response f(t).

• (b) Is it possible to transmit data without ISI?

• (c) How do the magnitudes of the tails of the overall impulse response decay with large values of t?

• (d) Suppose that binary signaling is used with this pulse shape so that the noiseless signal at the output of the receive filter is

  $$\displaystyle{y(t) =\sum _{n}x_{n}f(t - nT)}$$

  where x _n  ∈ {−1, +1}. What is the maximum possible magnitude that y(t) can achieve?

7.3. Suppose a digital communication system operates over an “ideal channel” and employs an overall pulse p(t) that has the Gaussian-shaped form

$$\displaystyle{p(t) = e^{-\pi a^{2}t^{2} }.}$$

• (a) Explain why p(t) does not admit intersymbol interference (ISI) free transmission.

• (b) To reduce the level of ISI to a relatively small amount, the condition that p(T) = 0. 01 is imposed, where T is the symbol interval. The bandwidth W of the pulse p(t) is defined as that value of W for which P(W)∕P(0) = 0. 01, where P(f) is the Fourier transform of p(t). Determine the value W and compare this value to that of a raised cosine spectrum with 100% roll-off.

7.4. Show that the ISI coefficients {f _n } may be expressed in terms of the channel vector coefficients {g _n } as

$$\displaystyle{f_{n} =\sum _{ k=0}^{L-n}g_{ k}^{{\ast}}g_{ k+n}\hspace{36.135pt} n = 0,1,2,\ldots,L.}$$

7.5. Suppose that BPSK is used on a static ISI channel. The complex envelope has the form

$$\displaystyle{\tilde{s}(t) = A\sum _{k=-\infty }^{\infty }x_{ k}h_{a}(t - kT),}$$

where x _k  ∈ {−1, +1} and h _a (t) is the amplitude shaping pulse. The full response rectangular pulse h _a (t) = u _T (t) is used and the impulse response of the channel is

$$\displaystyle{g(t) = g_{0}\delta (t) - g_{1}\delta (t-\tau ),}$$

where g ₀ and g ₁ are complex numbers and 0 < τ < T.

• (a) Find the received pulse h(t).

• (b) What is the filter matched to h(t)?

• (c) What are the ISI coefficients {f _i }?

7.6. Suppose that BPSK signaling is used on a static ISI channel having impulse response

$$\displaystyle{g(t) =\delta (t) + 0.1\delta (t - T).}$$

The receiver employs a filter that is matched to the transmitted pulse h _a (t), and the sampled outputs of the matched filter are

$$\displaystyle{y_{n} = x_{n}q_{0} +\sum _{k\neq n}x_{k}q_{n-k} +\eta _{n},}$$

where x _n  ∈ {−1, +1}. Decisions are made on the {y _n } without any equalization.

• (a) What is the variance of noise term η _n ?

• (b) What are the values of the {q _n }?

• (c) What is the probability of error in terms of the average received bit-energy-to-noise ratio?

7.7. A typical receiver for digital signaling on an ISI channel consists of a matched filter followed by an equalizer. The matched filter is designed to minimize the effect of random noise, while the equalizer is designed to minimize the effect of ISI. By using mathematical arguments, show that (1) the matched filter tends to accentuate the effect of ISI, and (2) the equalizer tends to accentuate the effect of random noise.

7.8. Consider an ISI channel, where f _n  = 0 for | n | > 1. Suppose that the receiver uses a filter matched to the received pulse h(t) = h _a (t) ∗ g(t), and the T-spaced samples at the output of the matched filter, {y _k }, are filtered as shown in Fig. 7.24. The values of g ₀ and g ₁ are chosen to satisfy

$$\displaystyle\begin{array}{rcl} \vert g_{0}\vert ^{2}\ +\ \vert g_{ 1}\vert ^{2}\ & =& \ f_{ 0} {}\\ g_{0}g_{1}^{{\ast}}\ & =& \ f_{ 1}. {}\\ \end{array}$$

Find an expression for the filter output v _k in terms of g ₀, g ₁, x _k , x _k−1, and the noise component at the output of the digital filter, η _k .

Fig. 7.24

Digital filter for Problem 7.8

7.9. The z-transform of the channel vector g of a communication system is equal to

$$\displaystyle{G(z) = 0.1 + 1.0z^{-1} - 0.1z^{-2}.}$$

A binary sequence x is transmitted, where x _k  = ∈ {−1, +1}. The received samples at the output of the noise whitening filter are

$$\displaystyle{v_{n} =\sum _{ k=0}^{2}g_{ k}x_{n-k} +\eta _{n},}$$

where {η _n } is a white Gaussian noise sequence with variance σ _η ² = N _o .

• (a) Evaluate the probability of error if the demodulator ignores ISI.

• (b) Design a 3-tap zero-forcing equalizer for this system.

• (c) What is the response {v _k } for the input sequence

  $$\displaystyle{\{x_{k}\} = (-1)^{k},\ k = 0,1,2,3\ ?}$$

  What is the response at the output of the equalizer?

• (d) Evaluate the probability of error for the equalized channel.

7.10. Suppose that a system is characterized by the received pulse

$$\displaystyle{h(t) = \sqrt{2a}\mathrm{e}^{-at},\hspace{36.135pt} 0 \leq t \leq \infty.}$$

A receiver implements a filter matched to h(t) and generates T-spaced samples at the output of the filter. Note that the matched filter is actually noncausal.

• (a) Find the ISI coefficients f _i .

• (b) What is the transfer function of the noise whitening filter that yields a system having an overall minimum phase response?

• (c) Find the transfer function of the equivalent zero-forcing equalizer C ^′(z).

• (d) Find the noise power at the output of the zero-forcing equalizer, and find the condition when the noise power becomes infinite.

7.11. Consider M-PAM on a static ISI channel, where the receiver employs a filter that is matched to the received pulse. The sampled outputs of the matched filter are

$$\displaystyle{y_{n} = x_{n}f_{0} +\sum _{k\neq n}x_{k}f_{n-k} +\nu _{n},}$$

where the source symbols are from the set { ± 1,  ± 3,  …,  ± (M − 1)}. Decisions are made on the {y _n } without any equalization by using a threshold detector. The ℓth ISI pattern can be written as

$$\displaystyle{D(\ell) =\sum _{k\neq n}x_{\ell,k}f_{n-k}}$$

and D(ℓ) is maximum when sgn(x _ℓ, k ) = sgn(f _n−k ) and each of the x _ℓ, k takes on the maximum signaling level, i.e., x _ℓ, k  = (M − 1)d for M even. The maximum distortion is defined as

$$\displaystyle{D_{\mathrm{max}} = \frac{1} {f_{0}}\sum _{n\neq 0}\vert f_{n}\vert.}$$

• (a) Discuss and compare error performance M-ary signaling (M > 2) with binary signaling (M = 2), using D _max as a parameter.

• (b) Suppose that the channel has ISI coefficients

  $$\displaystyle\begin{array}{rcl} f_{i}& =& 0.0\,\ \ \vert i\vert \geq 3 {}\\ f_{2}& =& f_{-2} = 0.1 {}\\ f_{1}& =& f_{-1} = -0.2 {}\\ f_{0}& =& 1.0. {}\\ \end{array}$$

  Plot the probability of error against the signal-to-noise ratio and compare with the ideal channel case, i.e., f ₀ = δ _n0. Show your results for M = 2 and 4.

7.12. Consider a linear MSE equalizer and suppose that the tap gain vector c satisfies

$$\displaystyle{\mathbf{c} = \mathbf{c}_{\mathrm{op}} + \mathbf{c}_{e}}$$

where c _e is the tap gain error vector. Show that the mean square error that is achieved with the tap gain vector c is

$$\displaystyle{J = J_{\mathrm{min}} + \mathbf{c}_{e}^{T }\mathbf{M}_{v}\mathbf{c}_{e}^{{\ast}}.}$$

7.13. The matrix M _v has an eigenvalue λ _k and eigenvector x _k if

$$\displaystyle{\mathbf{x}_{k}\mathbf{M}_{v} =\lambda _{k}\mathbf{x}_{k}\,\ \ k = 1,\ldots,N.}$$

Prove that the eigenvectors are orthogonal, i.e., x _i x _j ^T = δ _ij .

7.14. Show that the relationship between the output SNR and J _min for an infinite-tap mean-square error linear equalizer is

$$\displaystyle{\gamma _{\infty } = \frac{1 - J_{\mathrm{min}}} {J_{\mathrm{min}}},}$$

where γ _∞ indicates that the equalizer has an infinite number of taps. Note that this relationship between γ _∞ and J _min holds when there is residual ISI in addition to the noise.

7.15. In this question, it will be shown in steps that

$$\displaystyle{\nabla _{\mathbf{c}}J = 2\mathbf{c}^{T}\mathbf{M}_{ v} - 2\mathbf{v}_{x}^{H}.}$$

Define

$$\displaystyle\begin{array}{rcl} \mathbf{M}_{v}& \stackrel{\bigtriangleup }{=}& \mathbf{M}_{v_{R}} + j\mathbf{M}_{v_{I}} {}\\ \mathbf{c}& \stackrel{\bigtriangleup }{=}& \mathbf{c}_{R} + j\mathbf{c}_{I} {}\\ \mathbf{v}_{x}& \stackrel{\bigtriangleup }{=}& \mathbf{v}_{x_{R}} + j\mathbf{v}_{x_{I}}. {}\\ \end{array}$$

• (a) By using the Hermitian property M _v  = M _v ^H show that

  $$\displaystyle{\mathbf{M}_{v_{R}} = \mathbf{M}_{v_{R}}^{T}\mbox{ and }\mathbf{M}_{ v_{I}} = -\mathbf{M}_{v_{I}}^{T}.}$$

• (b) Show that

  $$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{c}_{R}}\mathrm{Re}\{\mathbf{v}_{x}^{H}\mathbf{c}^{{\ast}}\}& =& \mathbf{v}_{ x_{R}}^{T} {}\\ \nabla _{\mathbf{c}_{I}}\mathrm{Re}\{\mathbf{v}_{x}^{H}\mathbf{c}^{{\ast}}\}& =& -\mathbf{v}_{ x_{I}}^{T} {}\\ \nabla _{\mathbf{c}_{R}}\mathbf{c}^{T}\mathbf{M}_{ v}\mathbf{c}^{{\ast}}& =& 2\mathbf{c}_{ R}^{T}\mathbf{M}_{ v_{R}} - 2\mathbf{c}_{I}^{T}\mathbf{M}_{ v_{I}} {}\\ \nabla _{\mathbf{c}_{I}}\mathbf{c}^{T}\mathbf{M}_{ v}\mathbf{c}^{{\ast}}& =& 2\mathbf{c}_{ I}^{T}\mathbf{M}_{ v_{R}} + 2\mathbf{c}_{R}^{T}\mathbf{M}_{ v_{I}} {}\\ \end{array}$$

  where ∇ _x is the gradient with respect to vector x.

• (c) By defining the gradient of a real-valued function with respect to a complex vector c as

  $$\displaystyle{\nabla _{\mathbf{c}} = \nabla _{\mathbf{c}_{R}} + j\nabla _{\mathbf{c}_{I}}}$$

  show that

  $$\displaystyle\begin{array}{rcl} \nabla _{\mathbf{c}}\mathrm{Re}\{\mathbf{v}_{x}^{H}\mathbf{c}^{{\ast}}\}& =& \mathbf{v}_{ x}^{H} {}\\ \nabla _{\mathbf{c}}\mathbf{c}^{H}\mathbf{M}_{ v}\mathbf{c}^{{\ast}}& =& 2\mathbf{c}^{T}\mathbf{M}_{ v}. {}\\ \end{array}$$

7.16. Show that the pairwise error probability for digital signaling on an ISI channel is given by (7.171).

7.17. Consider the transmission of the binary sequence x, x _n  ∈ {−1, +1} over the equivalent discrete-time white noise channel model shown in Fig. 7.25. The received sequence is

$$\displaystyle\begin{array}{rcl} v_{0}& =& 0.70x_{0} +\eta _{1} {}\\ v_{1}& =& 0.70x_{1} - 0.60x_{0} +\eta _{2} {}\\ v_{2}& =& 0.70x_{2} - 0.60x_{1} +\eta _{3} {}\\ \vdots& & {}\\ v_{k}& =& 0.70x_{k} - 0.60x_{k-1} +\eta _{k}. {}\\ \end{array}$$

Fig. 7.25

Discrete-time white noise channel model for Problem 7.17

• (a) Draw the state diagram for this system.

• (b) Draw the trellis diagram.

• (c) Suppose that the received sequence is

  $$\displaystyle{\{v_{i}\}_{i=0}^{6} =\{ 1.0,\ -1.5,\ 0.0,\ 1.5,\ 0.0,\ -1.5,\ 1.0\}.}$$

  Show the surviving paths and their associated path metrics after v ₆ have been received.

7.18. Suppose that BPSK signaling is used on a frequency-selective fading channel. The discrete-time system consisting of the transmit filter, channel, receiver filter, and baud-rate sampler can be described by the polynomial

$$\displaystyle{F(z) = \frac{5} {16} -\frac{1} {8}z^{-1} -\frac{1} {8}z.}$$

The samples at the output of the receiver filter are processed by a noise whitening filter such that the overall discrete-time white noise channel model G(z) has minimum phase.

• (a) Find G(z).

• (b) Draw the state diagram and the trellis diagram for the discrete-time white noise channel model.

• (c) A block of 10 symbols x = { x _i } _i = 0 ⁹ is transmitted over the channel and it is known that x ₉ = −1. Assume that x _i  = 0, i < 0 and suppose that the sampled sequence at the output of the noise whitening filter is

  $$\displaystyle\begin{array}{rcl} \mathbf{v}& =& \{v_{0},v_{1},v_{2},v_{3},\ldots v_{9}\} {}\\ & =& \{1/2,1/4,-3/4,3/4,-3/4,-1/4,3/4,-3/4,-1/4,-1/4\}. {}\\ \end{array}$$

  What sequence x was most likely transmitted?