CE - Lect 4

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M. Junaid Mughal Fall 2010

Communication Electronics

Lecture 3: FYP Presentations

& QUIZ 1

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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.

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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

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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

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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

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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

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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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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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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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Waveform

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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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Waveform

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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

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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

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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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M Junaid Mughal Fall 2010

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

CE - Lect 4

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