Fourier series and transforms
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| Fourier transform | biến đổi Fourier |
| spectrum/ˈspektrəm/ | spectrum |
| brick-wall filter | brick-wall filter |
The bridge to frequency
- The Fourier transform $X(\omega) = \int x(t)e^{-j\omega t}\,dt$ carries a signal to its spectrum; convolution in time becomes multiplication in frequency: $Y = XH$.
A pulse of width $\tau$ transforms to $A\tau\,\mathrm{sinc}(\omega\tau/2)$: value $A\tau$ at DC, first null at $f = 1/\tau$. A 1 ms click spans to 1 kHz; a 0.1 ms click to 10 kHz.
{exercise:uni-ee0631-sinc}
A 0.1 ms rectangular pulse has its first spectral null at —
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Null at f = 1/τ = 1/0.0001 = 10 kHz. Ten times shorter in time, ten times wider in spectrum.
Convolution in the time domain becomes, in the frequency domain, —
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Y(ω) = X(ω)H(ω) — the property the whole subject turns on, and the reason filters are designed in frequency.
Steepness costs length
- The ideal brick-wall filter has a sinc impulse response stretching infinitely in both directions — non-causal. Truncating it to ±5 ms ripples the passband (Gibbs): frequency steepness is paid in time-domain length.
{exercise:uni-ee0632-wall}
The ideal brick-wall low-pass cannot be built because its impulse response is —
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A rectangle in frequency transforms to a sinc in time, nonzero before t = 0 — the filter must respond before the input. Truncation trades the impossibility for Gibbs ripple.