DTFT, DFT, ZT, and FFT 525
Example 5.9
Verify the linearity property of DFT that states, in short, the following:
The sum of DFT[f 1 (n)] + DFT[f 2 (n)] + DFT[f 3 (n)] ... is equal to the DFT of the sums of
(f 1 (n) + f 2 (n) + f 3 (n) ...), for the following specifi c case:
DFT u n DFT u n DFT
n
un
n
[()]() cos ()
1
3
^73
DFT u nu n
n
un n
n
() () cos ()
1
3
7 1
3
for
16
where
f 1 (n) = u(n)
fn un
n
2 () ()
1
3
fn
n
3 ()^7 cos un()
3
FIGURE 5.38
Plots of DFT of signal(t) = noise + f(t) of Example 5.8.
0
− 4
− 2
0
Angle
2
4
500 1000 1500 2000
frequency in Hertz
2500 3000 3500 4000
0
0
50
100
Magnitude
150
500 1000 1500 2000
DFT Magnitude and Phase
frequency in Hertz
2500 3000 3500 4000