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8/8/2019 CE - Lect 4
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M. Junaid Mughal Fall 2010
Communication Electronics
Lecture 3: FYP Presentations
& QUIZ 1
Umair Hashmi
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Communication Electronics
Lecture 4: Signal Analysis andMixing
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In this Lecture
Signal Analysis
Complex Waves
Trigonometric Fourier Series Fourier Series of a rectangular Waveform
Power and Energy Spectra
Effects of band-limiting a Signal
Frequency Spectrum and Bandwidth
Linear Summing
Nonlinear Mixing
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Signal Analysis
Signal Analysis is essentially the mathematical analysis of
frequency, bandwidth, and voltage level of a signal.
Periodic waves can be represented as sum of sine and cosine
waves and can be analyzed in time domain and frequencydomain.
Umair Hashmi
v (t) = V sin(2ft + ) or v (t) = V cos(2ft + )
i(t) = I sin(2ft + ) or i (t) = I cos(2ft + )
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Signal Analysis
Time-Domain Analysis
A description of a signal with respect to time is called a time-
domain representation of that signal.
A standard oscilloscope is the commonly used time-domain
instrument where the display tube of the CRT is an amplitude-
versus-time representation of the signal.
The vertical deflection on the oscilloscope indicates the
magnitude at different time and the horizontal deflection is a
function of time.
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Signal Analysis
Frequency-Domain Analysis
A description of a signal with respect to frequency is called a
frequency-domain representation.
The commonly used interest for studying a signal in
frequency domain is Spectrum Analyzer.
The height of each line shows the magnitude of the
frequency it represents. A frequency-domain representation does not indicate the
shape of the waveform.
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Signal Analysis
Fig. 2.2 and Fig. 2.3
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Complex Waves
A complex wave is any periodic (repetitive) waveform that is
not a sinusoid such as a square wave, rectangular wave and a
triangular wave and can be represented as sums of sines or
cosines.
Fourier Series:
It is a mathematical tool that allows us to move back andforth between time and frequency domains. In general, a
Fourier series can be written for any periodic function as a
series of sum of trigonometric functions.
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Fourier Series
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Fourier Series
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Fourier Series
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Fourier Series
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Fourier Series
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Complex Waves
Umair Hashmi
Even Symmetry
The periodic waveform is symmetric about the vertical
axis
coefficients are 0. The series contains a dc and cosine
terms.
f (t) = f (-t)
FIGURE
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Complex Waves
Umair Hashmi
Odd Symmetry
The periodic waveform is symmetric about line midway
between the vertical axis and the negative horizontal axis
and passing through the origin
coefficients are 0. The series contains a dc and sine
terms.
f (t) = - f (-t)
FIGURE
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Complex Waves
Umair Hashmi
Half-Wave Symmetry
The periodic waveform is such that the waveform for first
half cycle ( t= 0 to t = T/2) repeats itself with the opposite
sign in the second half-cycle ( t = T/2 to t = T)
f (t) = [- f (T+t) ] / 2
FIGURE
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Complex Waves
Umair Hashmi
EXAMPLE 2.1
For the train of square waves
a) Determine the peak amplitudes and frequencies of the first five
harmonics
b) Draw the frequency spectrum
c) Calculate the total instantaneous voltage for several times and sketch
the time-domainwaveform.
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Frequency Spectrum and Bandwidth
The frequency spectrum of a waveform consists of all the
frequencies contained in the waveform and their respective
amplitudes plotted in the frequency domain. Frequency
spectrums can show absolute values of frequency-versus some
unit of measurement such as dB.
The Bandwidth of a frequency spectrum is the range of
frequencies contained in the spectrum. The bandwidth is
calculated by subtracting the lowest frequency from the highest
frequency.
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Fourier Series of a Rectangular
Waveform
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The Duty Cycle (DC) of a waveform is the ratio of the active
time of the pulse to the period of the waveform.
The Fourier series for a rectangular voltage waveform with
even symmetry is:
TDC =
TDC ( % )= x 100
V
Tv(t) = + [ + + +
2 V
T
sin x (cos wt)
x
sin 2x (cos 2wt)
2x
sin nx (cos nwt
x
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Fourier Series of a Rectangular
Waveform
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The dc component of the waveform is equal to:
The amplitude for the nth harmonic is given by:
TVo = V x or V x DC
2V
TVn = x
sin nx
nx
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Fourier Series of a Rectangular
Waveform
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The dc component is equal to the pulse amplitude times the
duty cycle.
There are 0-V components at frequency 1/ hertz and all
integer multiples of that frequency provided that T = n where
n = any odd integer
The amplitude-versus-frequency time envelope of the
spectrum take on the shape of a damped sine wave in which all
spectrum components in odd-numbered lobes are positive and
all spectrum components in even-numbered lobes are negative.
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Fourier Series of a Rectangular
Waveform
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Fourier Series of a Rectangular
Waveform
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Narrowing the pulse width produces a frequency spectrum
with more uniform amplitude.
For infinitely narrow pulses, the frequency spectrum
comprises of an infinite number of harmonically related
frequencies of equal amplitude.
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Fourier Series of a Rectangular
Waveform
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Fourier Series of a Rectangular
Waveform
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EXAMPLE 2.2
For the pulse shown in figure
a) Determine the dc component
b) Draw the peak amplitudes of the first 10 harmonics
c) Plot the (sinx) / x functiond) Sketch the frequency spectrum
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Effect of Band limiting on Signals
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Communication channel can be replaced by an ideal linear-
filter with a finite bandwidth.
Bandlimiting a signal changes the frequency content and the
shape of the waveform.
If sufficient bandlimiting is imposes, only the fundamental
frequency is left.
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Effect of Band limiting on Signals
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LINEAR SUMMING
Umair Hashmi
Linear Summing occurs when two or more signals combine
in a linear device such as a passive network or a small-signal
amplifier.
No new frequency is produced and the combined waveform is
simply the linear addition of the individual signals.
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LINEAR SUMMING
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Single-Input Frequency
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LINEAR SUMMING
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Multiple Input Frequencies
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NON-LINEAR MIXING
Umair Hashmi
Non-linear mixing occurs when two or more signals are
combined in a non-linear device such as a diode or large-signal
amplifier.
With non-linear mixing, the input signals combine in a non-
linear fashion and produce additional frequency components.
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NON-LINEAR MIXING
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Single Input Frequency
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NON-LINEAR MIXING
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Multiple Input Frequencies
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Assignments & Quiz No 2
U i H h i
ASSIGNMENT (Tomasi) { Due on Wednesday 27th October)
Example 2.3
Problem 2.5
Problem 2.6
QUIZ No 2
Wednesday 27th October (10:15 11:00)
XC 3