PROPERTIES OF FOURIER REPRESENTATIONS

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Section 3.8 PROPERTIES OF FOURIER REPRESENTATIONS
Time property Periodic (t,n) Nonperiodic (t,n)
C.T. (t) Fourier Series (FS) Four Transform (FT)
D.T. n Discrete-Time Fourier Series (DTFS) Discrete-Time Fourier Transform (DTFT)
Non-periodic (k,w)
(3.19)
(3.35)
(T period)
(3.20)
(3.36)
Periodic (k,W)
(3.10)
(3.31)
(3.32)
(N period)
(3.11)
Continuous (w, W)
Discretek
Freq. property
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• Linearity and symmetry

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Example 3.30, p255
Find the frequency-domain representation of z(t).

• Which type of freq.-domain representation?
• FT, FS, DTFT, DTFS ?

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Periodic signals, continuous time. Thus, FS.
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Symmetry
We will develop using continuous, non-periodic
signals. Results for other 2 cases may be
obtained in a similar way.
a) Assume
? Further assume
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? Further assume
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b) Assume

• Convolution Applied to non-periodic signals.

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Conclusion Convolution in time domain ?
Multiplication in freq. domain.
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From results of example 3.26, p264.
Recall that (Example 3.25, p244)
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The same convolution properties hold for
discrete-time, non-periodic signals.
Convolution properties for periodic (DT or CT)
and periodic with non-periodic signals will be
discussed in Chapter 4.
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• Differentiation and integration (Section
  3.11)

• Applicable to continuous functions time (t) or
  frequency (w or W)
• FT (t, w) and DFTF (W)

Differentiation in time
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Find FT of
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Find x(t) if
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If x(t) is periodic, frequency-domain
representation is Fourier Series (FS)
Differentiation in frequency
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Example 3.40, p275
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This differential equation (it has the same
mathematical form as (), and thus the functional
form of G(jw) is the same as that of g(t) ) has a
solution given as
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Constant c can be determined as
Integration

• In time applicable to FT and FS
• In frequency applicable to FT and DTFT

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For w0, this relationship is indeterminate. In
general,
Determine the Fourier transform of u(t).
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Problem 3.29, p279 Fund x(t), given
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Review Table 3.6. Commonly used properties.
Problem 3.22(b), p271.
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• Time and frequency shift

Time shift
Note Time shift ? phase shift in frequency
domain. Phase shift is a linear function of w.
Magnitude spectrum does not change.
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Table 3.7, p280
Example 3.41 Find Z(jw)
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Problem 3.23(a), p282
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Frequency shift
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• Note
• Frequency shift ? time signal multiplied by a
  complex sinusoid.
• Carrier modulation.

Table 3.8, p284
Example 3.42, p284 Find Z(jw).
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Example 3.43, p285
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• Multiplication

READ derivation on p291!
Inverse FT of
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Periodic convolution
- p296, CT, periodic signals
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- p297, DT, periodic signals

• Scaling

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Example 3.48, p300
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Example 3.49, p301
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• Time scaling
• Time shift
• Differentiation

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• Parsevals relationship

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Table 3.10, p304
Example 3.50, p304
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• Time-bandwidth product
• Compression in time domain ? expansion in
  frequency domain
• Bandwidth The extent of the signals
  significant contents. It is in
• general a vague definition as significant
  is not mathematically defined.
• In practice, definitions of bandwidth include
• absolute bandwidth
• x bandwidth
• first-null bandwidth.
• If we define

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• Duality

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Example 3.52, p308
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Problem 3.44, p309
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Table 3.11, p311