Transform-Domain Systems

10.1 Frequency Response (FR)

10.1.1 Sinusoids as Eigenfunctions

The impulse response（脈衝響應）$h[n]$ completely describes an LSI (linear shift-invariant) system: the output is always the convolution $y[n] = x[n] \circledast h[n] = \sum_{m} h[m]\,x[n-m]$.

Feed in a complex sinusoid（複數弦波）$x[n] = e^{j\omega_0 n}$ and watch the convolution factor:

The complex scale factor is the DTFT of the impulse response evaluated at the input frequency. Writing it in polar form $H(e^{j\omega}) = |H(e^{j\omega})|\,e^{j\theta(\omega)}$, the output is

10.1.2 Real Sinusoids

In practice signals are real. Decompose a real cosine into two complex sinusoids and apply the eigenfunction property to each:

For a real impulse response $h[n]$ the frequency response is conjugate-symmetric, $H(e^{-j\omega}) = H^{*}(e^{j\omega}) = |H(e^{j\omega})|e^{-j\theta(\omega)}$, so the two terms recombine into a real cosine:

10.1.3 Transient and Steady State

Most signals start at a finite time（訊號從有限時間開始）, e.g. a gated complex sinusoid $x[n] = e^{j\omega_0 n}\,u[n]$. What does gating do? Split the convolution sum:

• The steady-state term is identical to the response to a pure (eternal) sinusoid.
• The transient term is the DTFT of the impulse-response tail from time $n$ onwards — a consequence of gating.
• For an FIR system of length $N$ the tail is empty for $n \ge N$: the transient is exactly zero after $N$ samples.
• For any stable IIR system $h[n] \to 0$, so the transient decays to zero for large $n$.

10.1.4 Example: Moving-Average Filter

10.2 Transfer Function (TF)

10.2.1 From Difference Equation to H(z)

Alternatively, take the z-transform of $y[n] = h[n] \circledast x[n]$: convolution becomes multiplication, $Y(z) = H(z)X(z)$. The transfer function（轉移函數） $H(z)$ can be obtained two equivalent ways:

For an FIR filter with $h[n] = \{h_0, h_1, \dots, h_{M-1}\}$ the two views coincide trivially: $p_k = h_k$, $d_0 = 1$, and the difference equation is $y[n] = \sum_{k=0}^{M-1} h_k\,x[n-k]$.

10.2.2 Poles, Zeros, ROC, and Stability

10.2.3 From H(z) to Frequency Response

The DTFT is the z-transform evaluated on the unit circle（單位圓）:

Factor $H(z)$ into first-order terms and evaluate at $z = e^{j\omega}$:

Taking magnitude and argument term by term:

10.2.4 Geometric Interpretation of the FR

Each factor $(e^{j\omega} - \nu)$ is a vector drawn from the root $\nu$ (pole or zero) to the point $e^{j\omega}$ moving along the unit circle. The whole frequency response is a product/ratio of these vectors:

First-order building block: a single zero

The factor $H(e^{j\omega}) = 1 - re^{j\theta}e^{-j\omega}$ (one zero at $z = re^{j\theta}$) is the elementary building block — every rational response is a product/ratio of such terms. Its log magnitude, phase and group delay all concentrate their activity around $\omega = \theta$:

10.2.5 Filtering: Shaping the Magnitude Response

Filtering means separating information in frequency with a constructed $H(e^{j\omega})$: given $x[n] = A\cos(\omega_1 n) + B\cos(\omega_2 n)$ where only the $\omega_1$ part is wanted, build $|H(e^{j\omega_1})| \approx 1$ and $|H(e^{j\omega_2})| \approx 0$, so that $y[n] \approx A\cos(\omega_1 n + \theta(\omega_1))$.

10.3 Phase Delay and Group Delay

10.3.1 Phase Delay

For a sinusoidal input $x[n] = \cos(\omega_0 n)$ we found $y[n] = |H(e^{j\omega_0})|\cos(\omega_0 n + \theta(\omega_0))$. The phase shift can be re-read as a time shift by factoring $\omega_0$ out:

The minus sign makes a positive $\tau_p$ mean a delay (causal behavior): a falling phase corresponds to the carrier arriving later.

10.3.2 Group Delay and Modulated Carriers

Phase delay describes a single sinusoid. Real signals carry information in envelopes. Consider a modulated carrier（調變載波）:

The signal occupies two close frequencies $\omega_c \pm \omega_m$. Assume $|H(e^{j\omega})| \approx 1$ around $\omega_c$, and call the phases at the two side frequencies $\theta(\omega_c - \omega_m) = \theta_l$ and $\theta(\omega_c + \omega_m) = \theta_u$. The output recombines into

The average phase shifts the carrier; the difference shifts the envelope. If $\theta(\omega)$ is locally linear around $\omega_c$, i.e. $\theta(\omega_c + \Delta\omega) \approx \theta(\omega_c) + S\,\Delta\omega$ with slope $S = \frac{d\theta}{d\omega}\big|_{\omega_c}$, then:

• Carrier phase shift $\frac{\theta_u + \theta_l}{2} = \theta(\omega_c)$ → carrier delayed by $-\theta(\omega_c)/\omega_c = \tau_p(\omega_c)$ — the phase delay.
• Envelope phase shift $\frac{\theta_u - \theta_l}{2} = \omega_m S$ → envelope delayed by $-S$ — the group delay.

10.3.3 Phase Distortion

If $\theta(\omega)$ is not linear across the band occupied by $A[n]$, different envelope frequencies suffer different delays and the envelope is smeared — phase distortion（相位失真）. The lecture closes with a striking example: a signal made of pulse bursts at different carrier frequencies passes through a system whose magnitude is nearly flat but whose group delay differs strongly between the two carrier bands — at the output the bursts arrive at different times and overlap differently, even though "the magnitude response barely touched them".

10.4 System Classes in the Transform Domain

(Supplementary to the slides.) Three classical families of systems are defined directly by their pole-zero geometry, and together they organize everything in 10.2–10.3: all-pass systems carry only phase, minimum-phase systems carry the least possible phase for a given magnitude, and linear-phase FIR systems carry a pure delay phase.

10.4.1 All-Pass Systems

An all-pass system（全通系統） has $|H_{ap}(e^{j\omega})| = $ constant for all $\omega$. The construction: pair every pole $a$ with a zero at its conjugate reciprocal $1/a^{*}$:

On the unit circle the vector to the zero at $1/a^{*}$ is always proportional in length to the vector to the pole at $a$ — the ratio of lengths is constant, so the magnitude never moves. Only the phase (and hence the group delay) varies.

10.4.2 Minimum-Phase and Inverse Systems

A causal stable system is minimum-phase（最小相位） if all its zeros (as well as poles) are inside the unit circle. Then the inverse system $1/H(z)$ — whose poles are the zeros of $H(z)$ — is also causal and stable: minimum-phase systems are exactly the ones that can be undone by a realizable filter (e.g. channel equalization).

Reflecting a zero $\zeta$ to $1/\zeta^{*}$ multiplies the response by an all-pass factor, which leaves $|H(e^{j\omega})|$ unchanged. Hence many systems share one magnitude, and:

Every rational $H(z)$ is a minimum-phase system (carrying all of the magnitude) cascaded with an all-pass (carrying the excess phase). Among all systems with the same $|H(e^{j\omega})|$, the minimum-phase one has:

• the smallest (continuous) phase lag and the smallest group delay at every $\omega$;
• the most front-loaded impulse response: partial energy $\sum_{n=0}^{m}|h[n]|^2$ is maximal for every $m$ (minimum energy delay).

10.4.3 Generalized Linear-Phase FIR Systems

A system has (generalized) linear phase（線性相位） if $H(e^{j\omega}) = A(\omega)\,e^{-j(\alpha\omega - \beta)}$ with $A(\omega)$ real. Then $\tau_g(\omega) = \alpha$ is constant — no phase distortion, every envelope is delayed by the same $\alpha$ samples. For causal FIR filters of length $M$ this happens exactly when the impulse response is symmetric or antisymmetric:

The symmetry constrains the zero locations: $z^{-(M-1)}H(1/z) = \pm H(z)$, so if $\zeta$ is a zero, then $1/\zeta$, $\zeta^{*}$, $1/\zeta^{*}$ are all zeros. The zero constellation therefore consists of:

• reciprocal-conjugate quadruples off the circle and off the real axis ($\zeta, \zeta^{*}, 1/\zeta, 1/\zeta^{*}$);
• conjugate pairs on the unit circle (self-reciprocal);
• reciprocal pairs on the real axis ($r, 1/r$);
• possible single zeros at $z = \pm 1$ (their own reciprocal and conjugate).

10.5 Chapter Summary

Exercises

b = [0 1 -1.2 1];              % numerator  z^-1 - 1.2 z^-2 + z^-3
a = [1 -1.3 1.04 -0.222];      % denominator

[H, w]   = freqz(b, a, 1024);  % frequency response on [0, pi)
[gd, wg] = grpdelay(b, a, 1024);

figure;
subplot(3,1,1); plot(w/pi, 20*log10(abs(H)));
ylabel('dB'); title('Log magnitude'); grid on;
subplot(3,1,2); plot(w/pi, angle(H));
ylabel('radians'); title('Phase (principal value)'); grid on;
subplot(3,1,3); plot(wg/pi, gd);
ylabel('samples'); xlabel('\omega/\pi'); title('Group delay'); grid on;

figure; zplane(b, a);          % poles x, zeros o
% roots(b(2:end)) -> zeros 0.6 +/- 0.8i   (|z| = 1, angle 0.2952*pi)
% roots(a)        -> poles 0.5 +/- 0.7i, 0.3 (|z| = 0.86, angle 0.3026*pi)