Analysis of Discrete-Time Systems

Chapter 9 argued that physical discrete-time systems arrive as difference equations. A digital filter is specified by a recursion, a control loop by an update rule, an economic model by a relation between this quarter's output and the last two — always the same shape, a linear constant-coefficient difference equation relating the output to its own past values and to the input:

Chapter 9 solved such equations by finding characteristic roots and matching a particular solution, which works but is laborious and keeps the natural and forced parts of the answer in separate compartments. The z-transform does the whole job at once, and it does it with two properties from Chapter 24 and nothing else: linearity, and the fact that a delay of \(k\) samples multiplies the transform by \(z^{-k}\).

Transform both sides. Every term on the left is a delayed copy of \(y\), so it contributes \(a_k z^{-k}Y(z)\); every term on the right contributes \(b_m z^{-m}X(z)\). Assume for the moment that the system starts from rest, so that the bilateral transform applies with no boundary terms — Section 25-4 removes that assumption.

An equation involving \(N+M+2\) delayed sequences has become a ratio of two polynomials whose coefficients are read straight off the recursion. This ratio is the system function, and it is the discrete-time counterpart of the transfer function of Chapter 19.

It is also, and this is the point that ties the chapter together, the z-transform of the impulse response. Setting \(x[n]=\delta[n]\) makes \(X(z)=1\), so \(Y(z)=H(z)\) and \(y[n]=h[n]\). The convolution property of Chapter 24 says the same thing from the other direction: \(y[n]=x[n]*h[n]\) transforms to \(Y(z)=X(z)H(z)\), so the ratio \(Y/X\) is \(H(z)\) whatever the input. The difference-equation description and the impulse-response description of Chapter 7 are two views of one object.

Factor the two polynomials. Writing the roots of the numerator as \(c_1,\dots,c_M\) and those of the denominator as \(d_1,\dots,d_N\),

The \(c_m\) are the zeros and the \(d_k\) are the poles. Suppose for now that the poles are distinct and that \(M \lt N\), so the ratio is strictly proper. A partial-fraction expansion in the variable \(z^{-1}\) then splits \(H\) into a sum of first-order terms, and each of those is a standard pair from Chapter 24.

Every pole contributes one term \(d_k^{\,n}\), which Chapter 9 called a natural mode. Nothing in the impulse response comes from anywhere else: the zeros do not create modes, they only fix the coefficients \(A_k\) with which the modes are mixed, and it is entirely possible for a zero to set some \(A_k\) to zero and silence a mode altogether.

Because the pole \(d_k\) is in general complex, write it as \(d_k = r_k e^{j\theta_k}\) and the behaviour of its mode becomes transparent: \(d_k^{\,n} = r_k^{\,n}e^{j\theta_k n}\). The magnitude of the pole controls growth or decay, and the angle of the pole controls oscillation. That single sentence is what a pole–zero plot is for.

Two refinements complete the picture. First, if the coefficients \(a_k, b_m\) are real — and for any physically realisable system they are — then Chapter 24's conjugation property forces complex poles to occur in conjugate pairs. Combining a pair \(re^{\pm j\theta}\) with conjugate coefficients gives a single real term of the form \(2|A|r^{n}\cos(\theta n + \angle A)\): a decaying oscillation whose frequency is the pole angle. A resonance therefore always costs two poles, never one.

Second, a pole repeated \(q\) times contributes not \(d^n\) alone but the family \(d^n, nd^n, \dots, n^{q-1}d^n\), by the z-differentiation property. The polynomial factor grows, but as long as \(|d| \lt 1\) the exponential wins and the mode still decays — slowly, and after an initial rise.

Finally, notice the extreme case \(N=0\), where the denominator is the constant \(a_0\) and there is no feedback at all. Then \(H(z)\) is a polynomial in \(z^{-1}\), all its poles sit at the origin, and \(h[n] = b_n/a_0\) for \(0\le n\le M\) and zero otherwise. The impulse response has finite length: this is an FIR system. As soon as any \(a_k\) with \(k\ge1\) is non-zero the recursion feeds output back into itself, the impulse response never terminates, and the system is IIR — the distinction Chapter 9 drew from the recursion, now visible as the presence or absence of poles away from the origin.

Chapter 8 defined the two properties an engineer cares about most in terms of the impulse response: a system is causal if \(h[n]=0\) for \(n \lt 0\), and BIBO stable if \(\sum_n|h[n]| \lt \infty\). Both translate into statements about the region of convergence, and once translated they become questions about where points lie in a picture.

Take causality first. A sequence that vanishes for negative \(n\) has a transform containing only non-positive powers of \(z\), so nothing blows up as \(|z|\to\infty\); conversely, Chapter 23 showed that a right-sided sequence always has an ROC that is the exterior of a circle. For a rational \(H(z)\), the boundary of that exterior is the largest pole magnitude.

Stability is even more direct. Evaluate the defining sum on the unit circle, where \(|z|=1\) and therefore \(|z^{-n}|=1\) for every \(n\):

The left side converges exactly when the right side does. So the transform converges on \(|z|=1\) if and only if \(h[n]\) is absolutely summable — the ROC contains the unit circle if and only if the system is stable. This is also the statement that the DTFT of Chapter 16 exists, which is why a stable system has a frequency response and an unstable one does not.

The two pictures show the same algebraic \(H(z)\) with the same two poles, and they are different systems. With a pole outside the unit circle you may have causality or stability but not both, and no amount of algebra will change that — the choice is forced by the geometry. This is the discrete-time echo of Chapter 19's result for the Laplace transform, where the left half-plane played the role that the interior of the unit circle plays here. The mapping \(s \to z\) implicit in sampling, which Part 6 developed, carries \(\operatorname{Re}s \lt 0\) onto \(|z| \lt 1\).

For a system given as a difference equation there is no need to compute the roots at all if you only want a yes-or-no answer. For the common second-order denominator \(D(z)=z^2 + a_1 z + a_2\) with real coefficients, a short argument gives a test. Write \(D(z)=(z-p_1)(z-p_2)\). The product of the roots is \(a_2\), so \(|a_2| = |p_1||p_2| \lt 1\) is necessary. And if both roots lie inside the unit circle then \(D(1)\) and \(D(-1)\) are both positive: for real roots each factor \((1-p_i)\) and \((1+p_i)\) is positive, and for a complex pair \(D(1)=|1-p_1|^2\) and \(D(-1)=|1+p_1|^2\) are squared magnitudes. Those two conditions read \(1+a_1+a_2 \gt 0\) and \(1-a_1+a_2 \gt 0\), which combine into a single inequality.

Everything so far assumed initial rest. A real system, however, is often switched on with energy already stored — a filter with non-zero contents in its delay registers, an accumulator with a running total. Chapter 9 handled this by solving the homogeneous equation separately and fixing the constants from the initial conditions. The unilateral z-transform does it in one pass, because the initial conditions enter the algebra automatically.

Recall from Section 24-3 that with \(X_u(z)=\sum_{n\ge0}x[n]z^{-n}\), a delay carries boundary terms:

Substituting these into the difference equation and solving for \(Y(z)\) always produces a sum of two rational functions. One carries the factor \(X(z)\) and vanishes if the input is zero; the other is built entirely from the stored values \(y[-1], y[-2],\dots\) and vanishes if the system starts at rest.

That last observation is worth pausing on, because it explains why Chapter 9's "natural plus forced" split and this "zero-input plus zero-state" split are not the same decomposition. The zero-state response contains natural modes too — they arrive through the partial-fraction expansion of \(H(z)X(z)\), whose poles include those of \(H\). The natural/forced split sorts terms by which pole they came from; the zero-input/zero-state split sorts them by which cause they came from. Both add up to the same \(y[n]\), and Worked Example 3 computes both.

One further point connects to Chapter 9's insistence on the initial-rest condition. A system with non-zero initial conditions is not LTI: doubling the input does not double the output, because the stored energy is unaffected. The zero-state part alone is LTI, and it is the part that \(H(z)\) describes. That is why the system function is a property of the system while the total response is not.

Chapter 16 defined the discrete-time Fourier transform and Chapter 15 used it to describe filtering. The link to the present chapter is a single substitution: when the ROC contains the unit circle, evaluating \(H(z)\) at \(z=e^{j\Omega}\) gives the frequency response.

As \(\Omega\) runs from \(0\) to \(2\pi\), the point \(e^{j\Omega}\) travels once anticlockwise round the unit circle, starting at \(z=1\) (which is DC), passing through \(z=j\) at \(\Omega=\pi/2\), reaching \(z=-1\) at \(\Omega=\pi\) — the highest discrete-time frequency, as Chapter 3 explained — and returning. The \(2\pi\)-periodicity of every discrete-time frequency response is now geometrically obvious: going round again visits the same points.

Multiply numerator and denominator by \(z^{N}\) to put the factored form in terms of \((z-c_m)\) and \((z-d_k)\). Taking magnitudes turns the product of complex numbers into a product of distances in the plane, and taking arguments turns it into a sum of angles.

The design rules follow without any further algebra. A pole at radius \(r\) and angle \(\theta\) produces a short denominator vector when \(\Omega\) is near \(\theta\), and the shorter that vector the larger the gain — so the response has a peak near \(\Omega=\theta\), and the peak becomes taller and narrower as \(r\to 1\). A zero on the unit circle at angle \(\phi\) makes the numerator vector vanish when \(\Omega=\phi\), so the response has an exact null there. Poles shape passbands, zeros carve stopbands, and the whole craft of filter design is placing them.

Chapter 8 established the interconnection rules in the time domain: systems in cascade convolve their impulse responses, systems in parallel add them. Under the convolution property those become multiplication and addition, which is why block-diagram algebra is done in the transform domain and never in the time domain.

Because multiplication commutes, a cascade may be reordered freely — a fact with real engineering consequences, since the ordering of sections changes the internal signal levels and therefore the arithmetic overflow behaviour of a fixed-point implementation, without changing the transfer function at all.

Feedback needs one line of algebra. Let the forward path be \(G(z)\) and the feedback path \(F(z)\), with the fed-back signal subtracted from the input. Writing \(E(z)\) for the error signal that actually drives the forward path,

The consequence deserves emphasis because it is the entire reason feedback is used. The open-loop system \(G(z)\) has whatever poles it has; closing the loop replaces them with the roots of \(1+GF=0\), which depend on the loop gain. Raising the gain drags those roots along continuous paths in the z-plane, and stability is retained only while every path stays inside the unit circle. Worked Example 5 computes the exact range of gain for which a simple loop remains stable, and the answer is a bounded interval — more gain is not better.

A system function is a specification; a structure is a way of computing it with adders, multipliers and delays. The same \(H(z)\) can be realised in many structures, all exact in infinite precision and all different once the arithmetic is finite.

The most literal realisation reads the difference equation aloud. Rearranging \(\sum_k a_k y[n-k] = \sum_m b_m x[n-m]\) with \(a_0=1\) gives the recursion \(y[n] = \sum_m b_m x[n-m] - \sum_{k\ge1}a_k y[n-k]\): compute a weighted sum of past inputs, subtract a weighted sum of past outputs. Drawn as a diagram this is Direct Form I, and it needs \(M\) delays for the input chain plus \(N\) for the output chain.

That is wasteful, and the convolution property says why. Direct Form I is the cascade of \(B(z)\) followed by \(1/A(z)\), and a cascade may be reordered. Put \(1/A(z)\) first, calling its output \(w[n]\), and both sections now delay the same signal — so one chain of delays can serve both.

This is Direct Form II, and it uses \(\max(M,N)\) delays instead of \(M+N\) — the minimum possible, which is why it is called canonical. Every delay is a word of memory in a real implementation, so the saving is not cosmetic.

Two further structures follow from the algebra of Section 25-2. Factoring \(H(z)\) into second-order sections and multiplying them gives the cascade realisation; expanding it in partial fractions and adding gives the parallel realisation. Both are preferred in practice over a single high-order direct form, because the sensitivity of a pole to a rounding error in a coefficient grows sharply with the order of the polynomial it belongs to. Splitting a tenth-order filter into five second-order sections keeps every coefficient error local.

Suppose a signal has been distorted by a known channel \(H(z)\) and you want it back. The natural request is a system \(H_i(z)\) such that the cascade of the two is the identity, \(h[n]*h_i[n] = \delta[n]\). Transforming, the requirement is simply \(H(z)H_i(z)=1\).

This exchange has a sharp consequence. For \(H_i\) to be causal and stable, all of its poles must lie inside the unit circle — that is, all the zeros of \(H\) must lie inside the unit circle. A system whose poles and zeros are all strictly inside the unit circle is called minimum phase, and it is exactly the class of systems that possess a causal, stable inverse.

If \(H\) has a zero outside the circle, you may still invert it, but you must give something up: a stable inverse exists but is not causal, and a causal inverse exists but is not stable. In practice one accepts a delay and approximates the non-causal inverse over a finite window, which is what an adaptive channel equaliser does.

The complementary object is the all-pass system, which has a pole at \(d\) and a zero at the reflected point \(1/d^*\). For any \(\Omega\) the two vectors from that pole and that zero to \(e^{j\Omega}\) have lengths in a constant ratio, so \(|H(e^{j\Omega})|\) is constant: the system changes phase alone. Any rational system can be written as a minimum-phase system in cascade with an all-pass one, which is how a troublesome zero outside the circle is quarantined — reflect it inside to get the minimum-phase part, and account for the difference with an all-pass factor that leaves the magnitude response untouched.

Problems 1 to 3 go from equation to response; 4 to 6 concern stability and frequency behaviour; 7 and 8 ask for reasoning rather than arithmetic. Sketch the pole–zero plot before computing anything — most of these can be half-answered from the picture.