Signals and Systems
Fall 2003
Lecture #10
7 October 2003
1,Examples of the DT Fourier Transform
2,Properties of the DT Fourier Transform
3,The Convolution Property and its
Implications and Uses
DT Fourier Transform Pair
– Analysis Equation
–FT
– Synthesis Equation
–Inverse FT
Convergence Issues
Synthesis Equation,None,since integrating over a finite interval
Analysis Equation,Need conditions analogous to CTFT,e.g.
— Absolutely summable
— Finite energy
Examples
Parallel with the CT examples in Lecture #8
More Examples
Infinite sum formula
Still More
4) DT Rectangular pulse
(Drawn for N
1
= 2)
5)
DTFTs of Sums of Complex Exponentials
Recall CT result:
What about DT:
Note,The integration in the synthesis equation is over 2π period,
only need X(e
jω
) in one 2π period,Thus,
a) We expect an impulse (of area 2π) at ω = ω
o
b) But X(e
jω
) must be periodic with period 2π
In fact
DTFT of Periodic Signals
Linearity
of DTFT
DTFS
synthesis eq.
Example #1,DT sine function
Example #2,DT periodic impulse train
— Also periodic impulse train – in the frequency domain!
Properties of the DT Fourier Transform
— Different from CTFT
More Properties
Example
— Important implications in DT because of periodicity
Still More Properties
Insert two zeros
in this example
(k=3)
But we can,slow” a DT signal down by inserting zeros:
k —an integer ≥ 1
x
(k)
[n] — insert (k - 1) zeros between successive values
Yet Still More Properties
7) Time Expansion
Recall CT property:
Time scale in CT is
infinitely fine
But in DT,x[n/2] makes no sense
x[2n] misses odd values of x[n]
Time Expansion (continued)
— Stretched by a factor
of k in time domain
-compressed by a factor
of k in frequency domain
Is There No End to These Properties?
8) Differentiation in Frequency
Total energy in
time domain
Total energy in
frequency domain
9) Parseval’s Relation
Differentiation
in frequency
Multiplication
by n
Example #1:
The Convolution Property
Example #2,Ideal Lowpass Filter
Example #3: