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
we may interpret



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.
$](http://www.dsprelated.com/josimages_new/mdft/img1534.png)

![$ \delta = [1,0,0,0]$](http://www.dsprelated.com/josimages_new/mdft/img1536.png)
The impulse signal is the identity element under convolution, since





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 , and implement the system by convolving the input
signal
with the impulse response
. 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
, the sampled frequency
response may be defined as


![$ [0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1128.png)







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









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



![$\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)}.
$](http://www.dsprelated.com/josimages_new/mdft/img1549.png)
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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Correlation Analysis
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