Title: Eeng 360 1
1 Chapter 2
- Fourier Transform and Spectra
- Topics
- Rectangular and Triangular Pulses
- Spectrum of Rectangular, Triangular Pulses
- Convolution
- Spectrum by Convolution
Huseyin Bilgekul Eeng360 Communication Systems
I Department of Electrical and Electronic
Engineering Eastern Mediterranean University
2Rectangular Pulses
3Triangular Pulses
4Spectrum of a Rectangular Pulse
- Rectangular pulse is a time window.
- FT is a Sa 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
5Spectrum of Sa Function
- To find the spectrum of a Sa function we can
use duality theorem. - Duality W(t) ?? w(-f)
Because ? is an even and real function
6Spectrum 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
7Spectrum of a Triangular Pulse
- The spectrum of a triangular pulse can be
obtained by direct evaluation of the FT integral. - An easier approach is to evaluate the FT using
the second derivative of the triangular pulse. - First derivative is composed of two rectangular
pulses as shown. - The second derivative consists of the three
impulses. - We can find the FT of the second derivative
easily and then calculate the FT of the
triangular pulse.
8Spectrum of a Triangular Pulse
9Spectrum of Rectangular and Sa Pulses
10Table 2.2 Some FT pairs
11Key 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)
12Convolution
- 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 convolution 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)).
13Example for Convolution
For 0lt t lt T
For t gt T
14Convolution
- 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(-2-t)
z(2-t)
z(0-t)
z(-6-t)
z(4-t)
t
2
0
1
-1
-6
-4
-2
y(t)
2
t
-4
0
1
-1
-2
-6