Eeng 360 1 - PowerPoint PPT Presentation

1 / 14
About This Presentation
Title:

Eeng 360 1

Description:

Chapter 2 Fourier Transform and Spectra Topics: Rectangular and Triangular Pulses Spectrum of Rectangular, Triangular Pulses Convolution Spectrum by Convolution – PowerPoint PPT presentation

Number of Views:111
Avg rating:3.0/5.0
Slides: 15
Provided by: PONCI
Category:
Tags: eeng | frequency | shift

less

Transcript and Presenter's Notes

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
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 ? 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.

Time Delay Theorem w(t-Td) ?? W(f) e-j?Td
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)h(t)
x(t)
h(t)
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(-?),
  • Time shifting of w2 by t seconds to obtain
    w2(-(?-t)), and
  • Multiplying this result by w1 to form the
    integrand w1(?)w2(-(?-t)).

13
Example for Convolution
For 0lt t lt T
For t gt T
14
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(-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
Write a Comment
User Comments (0)
About PowerShow.com