Appendix 2

1
Appendix 2

• Fourier Transform and Spectra
• Topics
• Rectangular and Triangular Pulses
• Spectrum of Rectangular, Triangular Pulses
• Convolution
• Spectrum by Convolution

2
Rectangular Pulses

• The sinc(.) function is defined as

3
Triangular Pulses
4
Spectrum of a Rectangular Pulse

• Rectangular pulse is a time window.
• FT is a sinc function, infinite frequency
  content.
• Shrinking time axis causes stretching of
  frequency axis.
• Signals cannot be both time-limited and
  bandwidth-limited.

Note the inverse relationship between the pulse
width T and the zero crossing 1/T
5
Spectrum of Sa Function
To find the spectrum of a Sa function we can use
duality theorem From Table 2.1 Duality W(t)
?? w(-f)
Because ? is an even and real function
graphically as
6
Spectrum of a Time Shifted Rectangular Pulse

• The spectra shown in previous slides are real
  because the time domain pulse ( rectangular
  pulse) is real and even.
• If the pulse is offset in time domain to destroy
  the even symmetry, the spectra will be complex.
• Let us now apply the Time delay theorem of Table
  2.1 to the Rectangular pulse.

Time Delay Theorem w(t-Td) ?? W(f) e-j?Td
We get
7
Spectrum of a Triangular Pulse
8
Spectrum of Rectangular and Sa Pulses
9
Table 2.2 Some FT pairs
10
Key FT Properties

• Time Scaling Contracting the time axis leads to
  an expansion of the frequency axis.
• Duality
• Symmetry between time and frequency domains.
• Reverse the pictures.
• Eliminates half the transform pairs.
• Frequency Shifting (Modulation) (multiplying a
  time signal by an exponential) leads to a
  frequency shift.
• Multiplication in Time
• Becomes complicated convolution in frequency.
• Mod/Demod often involves multiplication.
• Time windowing becomes frequency convolution with
  Sa.
• Convolution in Time
• Becomes multiplication in frequency.
• Defines output of LTI filters easier to analyze
  with FTs.

x(t)h(t)
x(t)
h(t)
11
Convolution

• The convolution of a waveform w1(t) with a
  waveform w2(t) to produce a third waveform w3(t)
  which is

where w1(t) w2(t) is a shorthand notation for
this integration operation and is read
convolved with. If discontinuous wave shapes
are to be convolved, it is usually easier to
evaluate the equivalent integral

• Evaluation of the integral involves 3 steps.
• Time reversal of w2 to obtain w2(-?),
• Time shifting of w2 by t seconds to obtain
  w2(-(?-t)), and
• Multiplying this result by w1 to form the
  integrand w1(?)w2(-(?-t)).

12
Example for Convolution
For 0lt t lt T
For t gt T
13
Convolution

• y(t)x(t)z(t) ?x(t)z(t- t )d t
• Flip one signal and drag it across the other
• Area under product at drag offset t is y(t).

z(t)
x(t)
z(t-t)
x(t)
z(t)
t
t
t
t
t
t
t1
t-1
0
1
-1
0
1
-1
z(0-t)
z(2-t)
z(-6-t)
z(-2-t)
z(-4-t)
z(4-t)
t
2
0
1
-1
-6
-4
-2
y(t)
2
t
-4
0
1
-1
-2
-6