Digital Signal Processing Reference
In-Depth Information
and
DFT (x
cO
[
]
=
[
]
n
)
Im
(X
k
)
(3.12)
Example 3.7.
Write a short script to verify Eqs. (3.11) and (3.12).
The following will suffice; different sequences can be substituted for
x
if desired.
x=[1:1:8]; subx = x(1,2:length(x)); xe = 0.5*(subx + fliplr(subx));
xo = 0.5*(subx - fliplr(subx)); xeven = [x(1), xe], xodd = [0, xo];
DF Tevenpt = fft(xeven), ReDF Tx = real(fft(x))
N=length(x); var = sum([(DF Tevenpt-ReDF Tx)/N].ˆ2)
3.9 GENERAL CONSIDERATIONS AND OBSERVATIONS
The computation of each bin value
X
[
]
is performed by doing a CZL (correlation at the zeroth lag)
between the signal sequence of length
N
and the complex correlator
k
]
Computation of the DFT using (3.7) directly is possible for smaller values of
N
. For larger values
of
N
,a
Fast Fourier Transform (FF T
) algorithm is usually employed. A subsequent section in this
chapter discusses the FFT.
cos
[
2
πkn/N
]−
j
sin
[
2
πnk/N
3.9.1 BIN VALUES
•If
x
[
n
]
is real-valued only, then
X
[
0
]
is real-valued only and represents the average or DC component
of the sequence
x
[
n
]
.
•If
x
is real-valued only, since the sine of the frequency
N/
2 is
identically zero. For odd-length DFTs, there is no Bin
N/
2.
[
n
]
is real-valued only,
X
[
N/
2
]
Example 3.8.
Compute the DFT of the sequence
[
1
,
1
]
.
Note that for a 2-point DFT, the only frequencies are DC and 1-cycle (which is the Nyquist rate,
N/
2). Both Bins 0 and 1 are therefore real. Bin 0 =
sum(([1,1]).*cos(2*pi*(0:1)*0/2))
= 2 (not scaling the
DFT by 1
/N
). For Bin 1, we get Bin 1 =
sum(([1,1]).*cos(2*pi*(0:1)*1/2))
= 0 (intuitively, you can see
that [1,1] is DC only, and has no one-cycle content).
Example 3.9.
Demonstrate the conjugate-symmetry property of the DFT of the real input sequence
cos
(
2*pi*1.3*(0:1:3)/4) + sin(2*pi*(0.85)*(0:1:3)/4)
We make the call
fft([cos(2*pi*1.3*(0:1:3)/4) + sin(2*pi*(0.85)*(0:1:3)/4)])
which returns the DFT as
