43 Properties of FT

1

• 4-3 Properties of FT
• 4-3-1 Linearity

• Example

2

• 4-3-2 Symmetry

• Proof (see whiteboard)

3

• Exercise E4.5 (p. 255)
• Want to show

4

• 4-3-3 Time Scaling

• Proof
• Time expansion implies frequency compression
  (see Fig. 4.18)
• In other words If f(t) is wider, then its
  spectrum is narrower, and vice versa

5

• Let a -1, then we have
• Time and Frequency Inversion (p. 256)

6

• 4-3-4 Time Shifting

• Proof
• This is to say Shifting in time results in a
  phase shift in frequency domain, but does not
  change the amplitude spectrum
• More questions

7

• 4-3-5 Frequency Shifting

• Proof

• Extension 1

• Extension 2

• The multiplication of by a sinusoid of
  frequency in the time domain results in a
  shifting by in the frequency domain

8

• This kind of multiplication is known as
  amplitude modulation
• The sinusoid is the carrier
• is the modulating signal
• is the modulated signal
• How to sketch? Observe that

• and act as envelopes for

9
sinc(t) clear pi 3.1415926 t
-2pi0.012pi ff sinc(t) plot(t,
ff) xlabel('t') ylabel('sinc(t)') grid
10
(No Transcript)
11
Multiplication clear pi 3.1415926 t
00.012pi f1 exp(-t) f2 sin(10t) ff
f1.f2 plot(t, f1, '--', t, f2, '-.', t,
ff) xlabel('t') ylabel('f(t)') title('Multiplic
ation of Two Signals') legend('exp(-t)',
'sin(10t)', 'exp(-t)sin(10t)', 1) grid
12
(No Transcript)
13

• Example 4.12

14
June 15, 2005 Example 4.12, Lathi clear w0
10 w -160.0116 f1 2sinc(2(ww0)) f
2 2sinc(2(w-w0)) plot(w, f1, w,
f2) xlabel('w') ylabel('F(w)') grid
15

• Reason 1
• Multiple signals are transmitted over the same
  media
• If all of them have the identical frequency,
  they will all interfere
• The receiver cannot get the expected information
• Therefore, we need to put these signals into
  different frequency bands
• This is called the frequency-division
  multiplexing (FDM)
• Reason 2
• The wavelength of voice signals is very large,
  leading to huge antennas

16

• 4-3-6 (a) Convolution in Time Domain

• Proof