Filters: Introduction

11.1 Filters and the Decibel Scale

11.1.1 What Is a Filter? FIR vs IIR

A digital filter（數位濾波器）is an LTI system designed to pass certain frequency components of a signal and attenuate（衰減）others. Because an LTI system simply multiplies each frequency component by its frequency response, the whole story of filtering is contained in

Frequencies where $|H(e^{j\omega})| \approx 1$ form the passband（通帶）; frequencies where $|H(e^{j\omega})| \approx 0$ form the stopband（阻帶）. This chapter introduces the simplest members of the two great families of realizable filters:

11.1.2 Decibels（分貝）

Filter gains span many orders of magnitude, so magnitude responses are usually plotted on a logarithmic decibel scale:

11.2 Simple FIR Filters

11.2.1 The Two-Point Average: a Simple FIR Lowpass

The simplest lowpass filter（低通濾波器）averages the current and previous samples:

Its frequency response factors into a real amplitude and a linear phase:

11.2.2 The M-Point Moving-Average Lowpass

Generalizing to an average of $M$ samples gives the moving-average filter（移動平均濾波器）, the workhorse of noise smoothing:

The magnitude is a periodic-sinc (Dirichlet) shape with nulls at $\omega = 2\pi k/M$, $k=1,\dots,M-1$: any sinusoid completing a whole number of cycles inside the averaging window averages to zero. Increasing $M$ narrows the passband and increases the smoothing, at the cost of a longer delay $(M-1)/2$ samples.

11.2.3 The First Difference: a Simple FIR Highpass

Flipping the sign of the second tap turns the averager into a differencer — the simplest highpass filter（高通濾波器）:

Now the zero sits at $z=+1$ ($\omega=0$): DC is annihilated, while the fastest alternation $(-1)^n$ passes with gain 1. Its 3-dB cutoff is again $\omega_c = \pi/2$. In fact $h_1[n]=(-1)^n h_0[n]$ — replacing $z$ by $-z$ shifts a frequency response by $\pi$, converting lowpass to highpass（把 $z\to-z$，頻率響應平移 π，低通變高通）.

11.3 Simple IIR Filters

Adding one feedback coefficient buys far more selectivity per multiply. All four classical responses — LP, HP, BP, BS — have standard first/second-order IIR prototypes, parameterized so that the gain is exactly 1 at the center of the passband and exactly 0 at the center of the stopband.

11.3.1 First-Order IIR Lowpass

The zero at $z=-1$ forces $H_{LP}(e^{j\pi})=0$; the normalizing constant forces $H_{LP}(e^{j0})=1$. The pole at $z=\alpha$ (real, $0<\alpha<1$ in practice) boosts low frequencies — the closer $\alpha$ is to 1, the narrower the passband. The 3-dB cutoff is controlled directly by $\alpha$:

11.3.2 First-Order IIR Highpass

Same pole, but the zero moves to $z=+1$: now DC is blocked and $\omega=\pi$ has unit gain. The same $\alpha \leftrightarrow \omega_c$ relation holds, so one design formula serves both filters.

11.3.3 Second-Order IIR Bandpass

A bandpass filter（帶通濾波器）needs zeros at both $\omega=0$ and $\omega=\pi$ and a resonant pole pair near the center frequency $\omega_0$:

The two parameters decouple beautifully: $\beta$ sets where the peak is ($|H_{BP}(e^{j\omega_0})|=1$ exactly), while $\alpha$ sets how narrow it is. Stability requires $|\alpha|<1$ and $|\beta|<1$.

11.3.4 Second-Order IIR Bandstop (Notch)

Now the zeros sit on the unit circle at $e^{\pm j\omega_0}$ with $\omega_0=\cos^{-1}\beta$, so the gain is exactly zero at the notch frequency（陷波頻率）— the classic tool for removing power-line hum（去除電源雜訊）. The same $\alpha$ formula gives the 3-dB notch width, and with matching parameters $H_{BP}(z)+H_{BS}(z)=1$: another doubly complementary pair.

11.4 Cascading Filters

A single first-order section rolls off slowly. The cheapest way to sharpen it is to cascade（串接）identical copies: overall $H(z)=G_1(z)G_2(z)\cdots G_K(z)$, magnitudes multiply, dB curves add.

Cascading $K$ copies of the two-point average gives $H(z)=\bigl[\tfrac12(1+z^{-1})\bigr]^K$ with magnitude $\cos^K(\omega/2)$. The stopband improves dramatically — at $\omega=3\pi/4$ one section gives −8.3 dB but four sections give −33 dB — yet there is a price: the passband shrinks too.

11.5 Ideal Filters and Practical Specifications

11.5.1 The Four Ideal Filter Types

An ideal filter（理想濾波器）has unit gain in the passband, zero gain in the stopband, and an instantaneous transition between them — a "brick wall"（磚牆式）response. The four classical types, on $0\le\omega\le\pi$ (responses are even in $\omega$ and $2\pi$-periodic):

• Lowpass (LP): pass $|\omega|\le\omega_c$, stop $\omega_c<|\omega|\le\pi$.
• Highpass (HP): stop $|\omega|<\omega_c$, pass $\omega_c\le|\omega|\le\pi$.
• Bandpass (BP): pass $\omega_1\le|\omega|\le\omega_2$ only（帶通）.
• Bandstop (BS): stop $\omega_1\le|\omega|\le\omega_2$ only（帶阻）.

11.5.2 The Ideal Lowpass Impulse Response — and Why It Is Unrealizable

Take the inverse DTFT of the ideal lowpass response $H_{LP}(e^{j\omega})=1$ for $|\omega|\le\omega_c$, $0$ otherwise:

11.6 Linear-Phase FIR Filters

11.6.1 Why Linear Phase? And Time-Reversal Filtering

Magnitude is only half the story. If the phase response $\angle H(e^{j\omega})$ is not linear in $\omega$, different frequency components experience different delays — the group delay（群延遲）$\tau(\omega)=-\,d\angle H/d\omega$ varies — and the waveform is smeared even when all magnitudes are untouched. This phase distortion（相位失真）is fatal in applications that care about waveform shape (ECG, seismic data, images, data pulses).

Every frequency is delayed by the same $D$ samples: the output is a filtered, shifted, but unsmeared copy of the input.

Time-reversal (zero-phase) filtering

For offline data there is an elegant trick. Filter the signal, time-reverse the result, filter again, and reverse once more. Each pass contributes $H(e^{j\omega})$ once with a conjugation, so the overall response is

— real and nonnegative: exactly zero phase, with the magnitude squared (doubling attenuation in dB). This is MATLAB's filtfilt. The catch: the procedure is noncausal, so it only works on recorded (finite, stored) signals, not in real time（僅適用於離線資料，無法即時處理）.

11.6.2 The Four Linear-Phase FIR Types

A causal length-$N$ FIR filter $h[n]$, $n=0,\dots,N-1$, has (generalized) linear phase if and only if its impulse response is symmetric or antisymmetric about its midpoint:

with $A(\omega)$ real (the amplitude response) and constant group delay $\tau=(N-1)/2$ samples. Two symmetry choices times two length parities give exactly four types:

11.6.3 Forced Zeros and Filter Suitability

The structural zeros in the table follow directly from the symmetry. Evaluate $H(z)=\sum h[n]z^{-n}$ at $z=\pm1$:

• Type 2 ($+$ symmetry, even $N$): at $z=-1$ the terms pair up as $h[n](-1)^n + h[N-1-n](-1)^{N-1-n}$; with $N-1$ odd the two exponents have opposite parity, so each pair cancels — $H(-1)=0$ always. A highpass or bandstop filter needs gain at $\omega=\pi$, so Type 2 cannot do it.
• Type 3 ($-$ symmetry, odd $N$): antisymmetry forces $H(1)=\sum h[n]=0$, and the parity argument also forces $H(-1)=0$. Both band edges are dead — only bandpass-like responses (and differentiators / Hilbert transformers（希爾伯特轉換器）, which want $A(0)=0$) are possible.
• Type 4 ($-$ symmetry, even $N$): only $H(1)=0$ is forced — fine for highpass, impossible for lowpass or bandstop.

11.7 Chapter Summary

Exercises

N = 200;  n = (0:N-1).';
s = 2*sin(2*pi*0.02*n);          % clean signal (f0 = 0.02 cycles/sample)
x = s + 0.8*randn(N,1);          % noisy observation

M = 11;
b = ones(1,M)/M;                 % moving-average coefficients
y = filter(b, 1, x);             % causal filtering, delay (M-1)/2 = 5
y_aligned = [y((M+1)/2:end); zeros((M-1)/2,1)];   % advance by 5 samples

subplot(2,1,1); plot(n, x, 'Color', [.5 .5 .5]); hold on;
plot(n, s, 'b', 'LineWidth', 1.5); legend('noisy x[n]','clean s[n]');
subplot(2,1,2); plot(n, y_aligned, 'r', 'LineWidth', 1.5); hold on;
plot(n, s, 'b--'); legend('filtered (aligned)','clean s[n]');

figure;                          % (b) magnitude response
[H, w] = freqz(b, 1, 1024);
plot(w/pi, abs(H)); grid on;
xlabel('\omega/\pi'); ylabel('|H(e^{j\omega})|');