Digital Signal Processing Reference
In-Depth Information
C
orrelation Value = 0
2
1
0
−1
0
20
40
60
80
100
120
(a) Cosine Corr Fcn (Circles) (Freq = 64) & Imp Resp (Stars); X−Axis = Sample
Correlation Value = 0
2
1
0
−1
0
20
40
60
80
100
120
(b) Sine Corr Fcn (Circles) (Freq = 64) & Imp Resp (Stars); X−Axis = Sample
0.015
0.01
0.005
0
0
10
20
30
40
50
60
(c) Bin Number of DFT (Bins 0 to N/2 Only)
Figure 3.44:
(a) Eight-sample impulse response (stars), and 128-point, 64-cycle cosine (real) correlator,
to compute the Real part of Bin 64; (b) Eight-sample impulse response (stars), and 128-point, 64-cycle
sine (imaginary) correlator, to compute the Imaginary part of Bin 64; (c) Magnitude of DFT, plotted up
to Bin 64.
N
−
1
X
[
k
]=
x
[
n
]
(
cos
[
2
πkn/N
−
j
sin
[
2
πkn/N
]
)
n
=
0
then the corresponding Inverse Discrete Fourier Transform (IDFT) is defined as
N
−
1
1
N
x
[
n
]=
X
[
k
]
(
cos
[
2
πkn/N
+
j
sin
[
2
πkn/N
]
)
(3.30)
k
=
0
where
n
runs from 0 to
N
- 1, as does
k
. Thus, the original time domain signal is reconstructed sample-
by-sample.
Example 3.39.
Compute the IDFT using the DFT coefficients of the sequence
[
2
,
1
,
−
1
,
1
]
.
The following code will obtain the DFT coefficients, initialize several values, then compute and
display the answer:
s = [2,1,-1,1]; n = 0:1:3;F=fft(s); k = [0:1:3]'; Arg = 2*pi*k/4;
for n = 0:1:3; hold on;
stem(n, real(0.25*sum(F(k+1)*exp(j*Arg*n)))); end
Perhaps a more intuitive way of thinking about the IDFT is to generate it not point-by-point, but
harmonic-by-harmonic. In this case,
