Signals and Systems
Fall 2003
Lecture #6
23 September 2003
1,CT Fourier series reprise,properties,and examples
2,DT Fourier series
3,DT Fourier series examples and
differences with CTFS
CT Fourier Series Pairs
Skip it in future
for shorthand
Another (important!) example,Periodic Impulse Train
— All components have:
(1) the same amplitude,
&
(2) the same phase.
(A few of the) Properties of CT Fourier Series
Linearity
Introduces a linear phase shift ∝ t
o
Conjugate Symmetry
Time shift
Example,Shift by half period
Parseval’s Relation
Energy is the same whether measured in the time-domain or the
frequency-domain
Multiplication Property
Periodic Convolution
x(t),y(t) periodic with period T
Periodic Convolution (continued)
Periodic convolution,Integrate over any one period (e.g,-T/2 to T/2)
Periodic Convolution (continued) Facts
1) z(t) is periodic with period T (why?)
2) Doesn’t matter what period over which we choose to integrate:
3)
Periodic
convolution
in time
Multiplication
in frequency!
Fourier Series Representation of DT Periodic Signals
x[n] - periodic with fundamental period N,fundamental frequency
Only e
jω n
which are periodic with period N will appear in the FS
So we could just use
However,it is often useful to allow the choice of N consecutive
values of k to be arbitrary.
There are only N distinct signals of this form
DT Fourier Series Representation
= Sum over any N consecutive values of k
k =<N>
∑
— This is a finite series
{a
k
} - Fourier (series) coefficients
Questions:
1) What DT periodic signals have such a representation?
2) How do we find a
k
Answer to Question #1:
Any DT periodic signal has a Fourier series representation
A More Direct Way to Solve for a
k
Finite geometric series
So,from
DT Fourier Series Pair
Note,It is convenient to think of a
k
as being defined for all
integers k,So:
1) a
k+N
= a
k
— Special property of DT Fourier Coefficients.
2) We only use N consecutive values of a
k
in the synthesis
equation,(Since x[n] is periodic,it is specified by N
numbers,either in the time or frequency domain)
Example #1,Sum of a pair of sinusoids
0
1/2
1/2
e
jπ/4
/2
e
-jπ/4
/2
0
0
a
-1+16
= a
-1
= 1/2
a
2+4×16
= a
2
= e
jπ/4
/2
Example #2,DT Square Wave
Using n = m - N
1
Example #2,DT Square wave (continued)
Convergence Issues for DT Fourier Series:
Not an issue,since all series are finite sums.
Properties of DT Fourier Series,Lots,just as with CT Fourier Series
Example: