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Signal and System Lecture 6

1) The document discusses discrete-time (DT) Fourier series, including their representation as a finite sum of complex exponentials with periodic Fourier coefficients. 2) Examples are provided to illustrate the DT Fourier series representation of periodic signals as sums of sinusoids, including a sum of two sinusoids and a square wave. 3) The key differences between continuous-time (CT) and DT Fourier series are noted, namely that DT Fourier series involve a finite number of terms due to the discrete and finite nature of digital signals.

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269 views19 pages

Signal and System Lecture 6

1) The document discusses discrete-time (DT) Fourier series, including their representation as a finite sum of complex exponentials with periodic Fourier coefficients. 2) Examples are provided to illustrate the DT Fourier series representation of periodic signals as sums of sinusoids, including a sum of two sinusoids and a square wave. 3) The key differences between continuous-time (CT) and DT Fourier series are noted, namely that DT Fourier series involve a finite number of terms due to the discrete and finite nature of digital signals.

Uploaded by

ali_rehman87
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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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 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
• Conjugate Symmetry

• Time shift

Introduces a linear phase shift ∝ to


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:

Periodic
3) convolution
in time

Multiplication
in frequency!
Fourier Series Representation of DT Periodic Signals
• x[n] - periodic with fundamental period N, fundamental frequency

• Only ejω 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.

⇓
DT Fourier Series Representation

∑ = Sum over any N consecutive values of k


k =<N >

— This is a finite series


{ak} - Fourier (series) coefficients

Questions:
1) What DT periodic signals have such a representation?

2) How do we find ak?


Answer to Question #1:
Any DT periodic signal has a Fourier series representation
A More Direct Way to Solve for ak

Finite geometric series


So, from
DT Fourier Series Pair

Note: It is convenient to think of ak as being defined for all


integers k. So:
1) ak+N = ak — Special property of DT Fourier Coefficients.

2) We only use N consecutive values of ak 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 a-1+16 = a-1 = 1/2


1/2
1/2 a2+4×16 = a2 = ejπ/4/2
ejπ/4/2
e-jπ/4/2
0
0
Example #2: DT Square Wave

Using n = m - N1
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:

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