Signals and Systems - Electrical Engineering

10.4 Discrete Fourier Transform 621

FIGURE 10.14
Computation of the FFT
of (a) a causal signal
and (b) a noncausal
signal. Notice that as
expected the magnitude
responses are
equal—only the phase
responses change.

(a)

(b)

x^1

[n

]

x[

n]

− 20 − 15 −^10 − 5 0 5 10 15 20

0

0.5

1

0

0.5

1

n

|X

( 1

je

ω)|

|X

(e

jω

)|

0

5

10

15

5

10

15

20

25

− 1

0

−0.5^0 0.5^1

ω/π

− 1 −0.5 0 0.5 1

ω/π

− 4

− 2

0

2

4

<X

( 1

je

ω)

−1 −0.5 0 0.5 1 1

ω/π

−1 −0.5 0 0.5

−4

−2

0

2

4

ω/π

<X

(e

jω

)

−10 −5 0 5 10 15 20 25 30 35 40

n

as the number of periods increases. Thus, we need to divide by the number of periods used in

computing the FFT.

Since the signal is sampled, it is of interest to have the frequency scale of the FFTs in hertz, so we

convert the discrete frequencyω(rad) intof(Hz) according to

f=

ω

2 πTs

=

ωfs

2 π