FIR filter design
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| FIR | Đáp ứng xung hữu hạn (FIR) |
| Hamming window | Hàm窗 Hamming |
| linear phase | linear phase |
The window decides everything
- An FIR filter truncates the ideal sinc with a window $w[n]$: the window's main lobe becomes the transition band, its sidelobes become the stopband. The Hamming window promises a $-43$ dB stopband and transition $\approx 3.3 f_s/N$.
50 Hz transition at $f_s = 1$ kHz: $N = 3.3\times 1000/50 = 66$ → take 67 taps (odd, Type-I), delay $(67-1)/2 = 33$ samples ✓.
{exercise:uni-ee1204-size}
A Hamming FIR at fs = 1 kHz must transition in 50 Hz. The rule N = 3.3·fs/Δf gives —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
N = 3.3×1000/50 = 66; choose the next odd number 67 so the symmetric Type-I delay (N−1)/2 = 33 is a whole number of samples.
Linear phase, then verify
- Symmetric taps $h[n] = h[N-1-n]$ give exactly linear phase — delay $(N-1)/2$ samples at every frequency, so waveforms keep their shape. Then measure: stopband at or below $-43$ dB and passband ripple inside ±0.05 dB, or the design is wrong.
{exercise:uni-ee1204-phase} {exercise:uni-ee1204-stopband}
Symmetric FIR taps give —
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h[n] = h[N−1−n] makes H a real function times a pure delay e^{−jω(N−1)/2}: every frequency is delayed equally.
The Hamming window's peak sidelobe — the stopband floor it promises — is —
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The window table sets the promise: rectangle −13 dB, Hann −31 dB, Hamming −43 dB, Blackman −58 dB — and the measurement must confirm it.