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Signals and SystemsFall 2003

Lecture #10

7 October 2003

1. Examples of the DT Fourier Transform

2. Properties of the DT Fourier Transform3. The Convolution Property and its

Implications and Uses

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DT Fourier Transform Pair

Analysis Equation

FT

Synthesis Equation

Inverse FT

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Convergence Issues

Synthesis Equation: None, since integrating over a finite interval

Analysis Equation: Need conditions analogous to CTFT, e.g.

Absolutely summable

Finite energy

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More Examples

Infinite sum formula

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Still More

4) DT Rectangular pulse (Drawn forN1 = 2)

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5)

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DTFTs of Sums of Complex Exponentials

Recall CT result:

What about DT:

Note: The integration in the synthesis equation is over 2 period,

only needX(ej) in one 2 period. Thus,

a) We expect an impulse (of area 2) at = ob) ButX(ej) must be periodic with period 2

In fact

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DTFT of Periodic Signals

Linearity

of DTFT

DTFS

synthesis eq.

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Example #1: DT sine function

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Example #2: DT periodic impulse train

Also periodic impulse train in the frequency domain!

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Properties of the DT Fourier Transform

Different from CTFT

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More Properties

Example

Important implications in DT because of periodicity

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Still More Properties

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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 ofx[n]

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Time Expansion (continued)

Stretched by a factor

ofkin time domain

-compressed by a factor

ofkin frequency domain

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Is There No End to These Properties?

8) Differentiation in Frequency

Total energy in

time domain

Total energy in

frequency domain

9) Parsevals Relation

Differentiationin frequency

Multiplicationby n

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Example #1:

The Convolution Property

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Example #2: Ideal Lowpass Filter

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Example #3: