1. Fourier Transform and Spectra
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

2. Rectangular Pulses

3. Triangular Pulses

4. Spectrum 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

5. Spectrum of Sa Function
To find the spectrum of a Sa function we can use duality theorem.
Duality: W(t)  w(-f)
Because Π is an even and real function

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. T 1
Time Delay Theorem: w(t-Td)  W(f) e-jωTd
We get:

7. Spectrum 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.

8. Spectrum of a Triangular Pulse

9. Spectrum of Rectangular and Sa Pulses

10. Table 2.2 Some FT pairs

11. 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)
x(t)*h(t)
h(t)
X(f)
X(f)H(f)
H(f)

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

13. Example for Convolution
For 0< t < T
For t > T

14. y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ
Convolution
y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ
Flip one signal and drag it across the other
Area under product at drag offset t is y(t).
x(t)
z(t)
z(t-t)
x(t)
z(t) t t t -1 1 t -1 1 t
t-1 t
t+1
z(-2-t)
z(-6-t)
z(0-t)
z(2-t)
z(4-t) -6 -4 -2 -1 1 2 t 2
y(t) -6 -4 -2 -1 1 2 t