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Filters and Convolution

A reason for the importance of convolution (defined in §7.2.4) is that every linear time-invariant system8.7can be represented by a convolution. Thus, in the convolution equation

$\displaystyle y = h\ast x \protect$ (8.1)

we may interpret $ x$ as the input signal to a filter, $ y$ as the output signal, and $ h$ as the digital filter, as shown in Fig.8.12.
Figure 8.12: The filter interpretation of convolution.
The impulse or ``unit pulse'' signal is defined by

$\displaystyle \delta(n) \isdef \left\{\begin{array}{ll}
1, & n=0 \\ [5pt]
0, & n\neq 0. \\
\end{array} \right.

For example, for sequences of length $ N=4$, $ \delta = [1,0,0,0]$. The impulse signal is the identity element under convolution, since

$\displaystyle (x\ast \delta)_n \isdef \sum_{m=0}^{N-1}x(m) \delta(n-m) = x(n).

If we set $ x=\delta$ in Eq.$ \,$(8.1) above, we get

$\displaystyle y = h\ast \delta = h.

Thus, $ h$, which we introduced as the convolution representation of a filter, has been shown to be more specifically the impulse response of the filter. It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response [68]. No matter what the LTI system is, we can feed it an impulse, record what comes out, call it $ h(n)$, and implement the system by convolving the input signal $ x$ with the impulse response $ h$. In other words, every LTI system has a convolution representation in terms of its impulse response.

Frequency Response

Definition: The frequency response of an LTI filter may be defined as the Fourier transform of its impulse response. In particular, for finite, discrete-time signals $ h\in{\bf C}^N$, the sampled frequency response may be defined as

$\displaystyle H(\omega_k) \isdef \hbox{\sc DFT}_k(h).

The complete (continuous) frequency response is defined using the DTFT (see §B.1), i.e.,

$\displaystyle H(\omega) \isdef \hbox{\sc DTFT}_\omega(\hbox{\sc ZeroPad}_\infty(h)) \isdef \sum_{n=0}^{N-1}h(n) e^{-j\omega n}

where the summation limits are truncated to $ [0,N-1]$ because $ h(n)$ is zero for $ n<0$ and $ n>N-1$. Thus, the DTFT can be obtained from the DFT by simply replacing $ \omega_k$ by $ \omega$, which corresponds to infinite zero-padding in the time domain. Recall from §7.2.10 that zero-padding in the time domain gives ideal interpolation of the frequency-domain samples $ H(\omega_k)$ (assuming the original DFT included all nonzero samples of $ h$).

Amplitude Response

Definition: The amplitude response of a filter is defined as the magnitude of the frequency response

$\displaystyle G(k) \isdef \left\vert H(\omega_k)\right\vert.

From the convolution theorem, we can see that the amplitude response $ G(k)$ is the gain of the filter at frequency $ \omega_k$, since

$\displaystyle \left\vert Y(\omega_k)\right\vert = \left\vert H(\omega_k)X(\omega_k)\right\vert
= G(k)\left\vert X(\omega_k)\right\vert,

where $ X(\omega_k)$ is the $ k$th sample of the DFT of the input signal $ x(n)$, and $ Y$ is the DFT of the output signal $ y$.

Phase Response

Definition: The phase response of a filter is defined as the phase of its frequency response:

$\displaystyle \Theta(k) \isdef \angle{H(\omega_k)}

From the convolution theorem, we can see that the phase response $ \Theta(k)$ is the phase-shift added by the filter to an input sinusoidal component at frequency $ \omega_k$, since

$\displaystyle \angle{Y(\omega_k)} = \angle{\left[H(\omega_k)X(\omega_k)\right]}...
... \angle{H(\omega_k)} + \angle{X(\omega_k)}
= \Theta(k) + \angle{X(\omega_k)}.

The topics touched upon in this section are developed more fully in the next book [68] in the music signal processing series mentioned in the preface.
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