Filter Functions

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Consider a signal (or random process) $y (t)$ , filtered to form a new function $u (t)$ , which is a linear functional of $y (t)$

$u(t) = \int_{-\infty}^{\infty} g(t-t') y(t') dt' \ ,$

where $g (τ)$ is the filter's kernel and depends only on the time difference $τ = t - t'$ for a stationary filter (which is assumed here). In what follows, it will be convenient to work with the Fourier transform of the kernel

$G(f) = \int_{-\infty}^{\infty} g(\tau) e^{2 i \pi f \tau} d\tau \ ,$

which carries the same information as $g (τ)$ . The usefulness of this quantity is readily apparent when we consider the convolution theorem of Fourier transforms: if the Fourier transform $Y(f) = \mathcal{F}\left[y(t)\right]$ exists, then the Fourier transform of the filter output $U(f) = \mathcal{F}\left[u(t)\right]$ is given by

$U(f) = G(f) Y(f) \ .$

It follows then, that if the input $y (t)$ is a random process with spectral density $S y (f)$ , then the spectral density of the filter output is given by:

$S_u(f) = \left|G(f)\right|^2 S_y(f) \ .$

It is often the case that the easiest way to determine the quantity $G (f)$ is to simply send $e 2 i π f t$ through the filter (rather than directly computing the Fourier transform):

$\int_{-\infty}^\infty g(t-t') e^{2 i \pi f t'} dt' = G^*(f) e^{2 i \pi f t} \ ,$

which differs from $G (f)$ by complex conjugation and a phase.

Bandwidth

The bandwidth of a filter is defined as

$\Delta f = \frac{\int^\infty_0 \left|G(f)\right|^2 df}{\left|G(f_0)\right|^2} \ ,$

where $f 0$ is the frequency at which $\left|G(f)\right|$ is maximal.

Some Examples

All pass filter: $u (t) = y (t)$

$g(\tau) = \delta(\tau) \quad , \quad G(f) = 1$

Differentiator: $u (t) = d y / d t$

$g(\tau) = \delta'(\tau) \quad , \quad G(f) = -2\pi i f$

Box Car Averager: $u(t) = \frac{1}{T}\int^t_{t-T} y(t') dt'$

$g(\tau) = \left\{\begin{matrix}1/T&,&T>\tau>0\\ 0&,&\textrm{else}\end{matrix}\right\} \quad , \quad G(f) = \frac{\sin{(\pi f T)}}{\pi f T}e^{i \pi f T} \quad , \quad \Delta f = 1/2T$

Finite Fourier Transform (bandpass filter): $u(t) = \frac{1}{T}\int^t_{t-T} \cos{\left[2\pi f_0(t-t')\right]}y(t') dt'$

$g(\tau) = \left\{\begin{matrix} \frac{1}{T} \cos{\left[2\pi f_0\tau\right]} &,& T>\tau>0 \\ 0&,&\textrm{else}\end{matrix}\right\} \quad , \quad G(f) = \frac{\sin{(\pi (f+f_0) T)}}{2\pi (f+f_0) T}e^{i \pi (f+f_0) T} + \frac{\sin{(\pi (f-f_0) T)}}{2 \pi (f-f_0) T}e^{i \pi (f-f_0) T} \quad , \quad \Delta f = 1/T$

Note that for $f_0 T \gg 1$ , the expression for $| G (f) |$ simplifies to

$|G(f)| \approx \left|\frac{\sin{(\pi (f-f_0) T)}}{2 \pi (f-f_0) T}\right|$

which is like the filter function for a box car averager, but with the frequency shifted and the peak magnitude smaller by $1 / 2$ . Furthermore, the introduction of the frequency shift $f 0$ means that the effective bandwidth is $\Delta f = \frac{1}{T}$ versus $\Delta f = \frac{1}{2T}$ for a box car averager. This discrepancy is why homodyne measurements can be "twice as accurate" as heterodyne measurements.

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Categories: Electronics | Concept

Filter Functions

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