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CHAPTER 3
The Discrete-time
Fourier Analysis
We have seen how a linear and time-invariant system can be represented
using its response to the unit sample sequence. This response, called the
unit impulse response h(n), allows us to compute the system response to
any arbitrary input x(n) using the linear convolution:
x(n) −→ h(n) −→ y(n) =h(n) ∗ x(n)
This convolution representation is based on the fact that any signal
can be represented by a linear combination of scaled and delayed unit
samples. Similarly, we can also represent any arbitrary discrete signal
as a linear combination of basis signals introduced in Chapter 2. Each
basis signal set provides a new signal representation. Each representation
has some advantages and some disadvantages depending upon the type
of system under consideration. However, when the system is linear and
time-invariant, only one representation stands out as the most useful. It
is based on the complex exponential signal set {e jωn } and is called the
discrete-time Fourier transform.
3.1 THE DISCRETE-TIME FOURIER TRANSFORM (DTFT)
|x(n)| < ∞, then its discrete-
If x(n)isabsolutely summable, that is, ∑ ∞
time Fourier transform is given by
∞∑
X(e jω ) = △ F[x(n)] =
−∞
n=−∞
x(n)e −jωn (3.1)
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