Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
2. Linearity
If
then
x1[ n]
↔ X
1[
k]
and
x
2[ n]
↔ X
2[
k]
Ax n]
+ Bx [ n]
↔ AX [ k]
+ BX [ ] (6.5)
1[ 2
1
2
k
where ↔ represents the pair of DFT and IDFT, and A and B are constants. This property includes an equal magnification
rule, and a superposition rule between the input and output.
3. Time-shifting
If x[ n]
↔ X [ k]
then
2πkn
− ↔ −
0
x[
n n0
] X [ k]exp
j (6.6)
N
2πkn
0
The time shifting will cause a change of spectrum in phase, not in the magnitude, because exp
− j = 1. .
N
4. Convolution
then
If x n]
↔ X [ ] and x n]
↔ X [ ]
1[ 1
k
2[ 2
k
N
1
∑ − x1[ n]
x
2[
m − n]
↔ X
1[
k]
X
2[
k]
m= 0
(6.7)
The relationship of convolution between two signals in time domain can be simplified to a multiplication in the frequency
domain. In the formula, the convolution is defined on one period.
5. Modulation
then
If
x1[ n]
↔ X
1[
k]
and
x
2[ n]
↔ X
2[
k]
x
N
− 1
1
[ n]
x2[
n]
↔ X
1[
m]
X
2[
k − m]
m=
0
(6.8)
Likewise to the property 4), the relationship of convolution between two spectra in the frequency domain can be simplified
to a multiplication in the time domain.
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