Digital Signal Processing Reference
In-Depth Information
IDTFS of
X(k)
with period
N
:
N
−
1
N
−
1
1
N
1
N
x(n)e
−
j(
2
π/N)kn
x(n)W
kn
,
X
p
(k)
=
=
−∞ ≤
k
≤∞
n
=
0
n
=
0
(3.89)
The discrete Fourier transform (DFT) and its inverse (IDFT) are a subset of
the DTFS and IDTFS coefficients, derived from the periodic DTFS and IDTFS
coefficients. They can be considered as nonperiodic sequences. A few examples
were worked out to show that the values of the DTFT, when evaluated at the
discrete frequencies, are the same as the DFT coefficients:
DFT of
x(n)
with length
N
:
N
−
1
N
−
1
x(n)e
−
j(
2
π/N)kn
x(n)W
kn
,
X(k)
=
=
0
≤
k
≤
(N
−
1
)
n
=
0
n
=
0
(3.90)
IDFT of
X(k)
with length
N
:
N
−
1
N
−
1
1
N
1
N
X(k)e
j(
2
π/N)kn
X(k)W
−
kn
,
x(n)
=
=
0
≤
n
≤
(N
−
1
)
k
=
0
k
=
0
(3.91)
The FFT algorithm for computing the DFT-IDFT coefficients offers very
significant computational efficiency and hence is used extensively in signal pro-
cessing, filter analysis, and design. It provides a unified computational approach to
find the frequency response from the time domain and vice versa. More examples
are added to show that the use of FFT and IFFT functions from MATLAB pro-
vides a common framework for getting the frequency response of a discrete-time
system from the discrete-time signal and finding the discrete-time signal from
the frequency response. Remember that the terms
discrete-time
(digital)
signal,
sequence
,or
function
have been used interchangeably in this topic; we have also
used the terms
discrete-time Fourier transform
(DTFT),
frequency response
,and
spectrum
synonymously in this chapter.
PROBLEMS
e
−
0
.
1
t
u(t)
is sampled to generate a DT signal
f(n)
at such
a high sampling rate that we can assume that there is no aliasing. Find a
closed-form expression for the frequency response of the sequence
f(n)
.
3.1
A signal
f(t)
=
te
−
0
.
1
t
u(t)
and
choose a frequency at which the attenuation is more than 60 dB. Assuming
3.2
Find the Fourier transform
X(jω)
of the signal
x(t)
=
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