Signals and Systems - Electrical Engineering

572 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

In this chapter, we will see that a great deal of the Fourier representation of discrete-time signals

and characterization of discrete systems can be obtained from our knowledge of the Z-transform.

To obtain the DFT, which is of great significance in digital signal processing, we will proceed in an

opposite direction as in the continuous-time analysis. First, we consider the Fourier representation of

aperiodic signals and then that of periodic discrete-time signals, and finally use this representation

to obtain the DFT.

10.2 Discrete-Time Fourier Transform

The discrete-time Fourier transform (DTFT) of a discrete-time signalx[n],

X(ejω)=

∑

n

x[n]e−jωn −π≤ω < π (10.1)

convertsx[n]into a functionX(ejω)of the discrete frequencyω(rad), while the inverse transform gives back

x[n]fromX(ejω)according to

x[n]=

1

2 π

∫π

−π

X(ejω)ejωndω (10.2)

Remarks

n The DTFT measures the frequency content of a discrete-time signal. When using the DTFT, it is important

to remember some of the differences between the continuous and the discrete domains. Discrete-time

signals are only defined for uniform sample times nTsor integers n, and the discrete frequency is such that

it repeats every 2 πradians (i.e.,ω=ω+ 2 πk for any integer k), so that X(ejω)is periodic and only the

frequencies[−π,π)need to be considered.

n The DTFT X(ejω)is periodic of period 2 π. Indeed, for an integer k,

X(ej(ω+^2 πk))=

∑

n

x[n]e−j(ω+^2 πk)n=X(ejω)

since e−j(ω+^2 πk)n=e−jωne−j^2 πkn=e−jωn. Thus, one can think of Equation (10.1) as the Fourier series

of X(ejω): Ifφ= 2 πis the period, the Fourier series coefficients are given by

x[n]=

1

φ

∫

φ

X(ejω)ej^2 πnω/φdω=

1

2 π

∫π

−π

X(ejω)ejnωdω

n For the DTFT to converge, as an infinite sum, it is necessary that

|X(ejω)|≤

∑

n

|x[n]||ejωn|=

∑

n

|x[n]|<∞