Signals and Systems

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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 anddifferences with CTFS
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CT Fourier Series Pairs
Review
Skip it in future for shorthand
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Another (important!) example Periodic Impulse
Train
Sampling function important for sampling

• - All components have
• the same amplitude,
• the same phase.

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(A few of the) Properties of CT Fourier Series
Linearity Conjugate Symmetry
Time shift
Introduces a linear phase shift ?to
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Example Shift by half period
using
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Parsevals Relation
Power in the kth harmonic
Average signal poser
Energy is the same whether measured in the
time-domain or the frequency-domain
Multiplication Property
(Both x(t) and y(t) are periodic with the same
period T)
Proof
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Periodic Convolutionx(t), y(t) periodic with
period T
Not very meaningful
E.g. If both x(t) and y(t) are positive, then
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Periodic Convolution (continued) Periodic
convolutionIntegrate over anyone period (e.g.
-T/2 to T/2)
where
otherwise
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Periodic Convolution (continued) Facts
1) z(t) is periodic with period T (why?)
2) Doesnt matter what period over which we
choose to integrate
Periodic 3) Convolution
in time
Multiplication In frequency!
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Fourier Series Representation of DT Periodic
Signals
xn -periodic with fundamental period N,
fundamental frequency
Only e j?n which are periodic with period N
will appear in the FS
There are only N distinct signals of this
form
So we could just use However, it is
often useful to allow the choice of N consecutive
values of k to be arbitrary.
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DT Fourier Series Representation
Sum over any N consecutive values of k This
is a finite series
- Fourier (series) coefficients
Questions 1) What DT periodic signals have
such a representation? 2) How do we find ak?
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Answer to Question 1 Any DT periodic signal has
a Fourier series representation
N equations for N unknowns, a0, a1, , a N-1
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A More Direct Way to Solve for ak Finite
geometric series
otherwise
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So, from
multiply both sides by
and then
orthoronality
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DT Fourier Series Pair
(Synthesis equation)
(Analysis equation)
NoteIt is convenient to think of akas being
defined for all integers k. So
1) akN ak Special property of DT Fourier
Coefficients. 2) We only use Nconsecutive
values of ak in the synthesis equation. (Since
xn is periodic, it is specified by N numbers,
either in the time or frequency domain)
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Example 1 Sum of a pair of sinusoids
periodic with period
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Example 2 DT Square Wave
multiple of
Using
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Example 2 DT Square Wave (continued)
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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
Frequency shift