Fourier series and transforms

Why does a click hit every band?

• A sharp click lights up every slider on a graphic equaliser at once, from bass to treble. A held note lights up one.
• The Fourier transform explains why: short events spread across many frequencies. This lesson builds the transform and its central trade-off.

From series to transform

• A periodic signal is a sum of harmonics. Let the period grow without limit and the harmonics merge into a continuous spectrum: the Fourier transform 傅里叶变换.
• $X(\omega) = \displaystyle\int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt$ and back again, $x(t) = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty} X(\omega)\,e^{j\omega t}\,d\omega$.

Convolution becomes multiplication

• The property the whole subject rests on: if $y = x * h$, then $Y(\omega) = X(\omega)H(\omega)$.
• $H(\omega)$ is the system's frequency response 频率响应. A filter is designed by shaping it, then multiplying, with no convolution integral.

The pulse and its sinc

• A pulse of height $A$ and width $\tau$, centred at 0, transforms to $X(\omega) = A\tau\,\dfrac{\sin(\omega\tau/2)}{\omega\tau/2}$, a sinc.
• At $\omega = 0$ it equals the area $A\tau$. The first null is at $\dfrac{\omega\tau}{2} = \pi$, which is $f = \dfrac{1}{\tau}$.

Time–bandwidth trade-off

• Shorter in time means wider in frequency. A 1 ms click has its first null at 1 kHz; a 0.1 ms click pushes it to 10 kHz.
• This time–bandwidth trade-off 时间带宽权衡 is unavoidable. That is why a click reaches every equaliser band.

Worked example: the ideal filter's price

• A brick-wall low-pass passing 0 to 4 kHz has a rectangular $H$. Its impulse response is a sinc in time, $h(t) = \dfrac{\sin(\omega_{c}t)}{\pi t}$, stretching to $\pm\infty$.
• It starts before $t = 0$, so it is non-causal and cannot be built. Cut it to $|t| \le 5$ ms and the edge softens to about $\dfrac{1}{10\,\text{ms}} = 100$ Hz wide, with Gibbs ripple.

Three common slips

Recap