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EE 23353 Analog Communications
Chapter 3: Analysis and Transmission of Signals
Dr. Rami A. Wahsheh
Communications Engineering Department
Chapter 3: Analysis and Transmission of Signals
3.1 Aperiodic signal representation by Fourier integral.
3.2 Transforms of some useful functions.
3.3 Some properties of the Fourier transform.
3.4 Signal transmission through a linear system.
3.5 Ideal and practical filters.
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3.6 Signal distortion over a communication channel.
3.7 Signal energy and energy spectral density.
3.8 Signal power and power spectral density.
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Fourier Transform
• The motivation for the Fourier transform comesfrom the study of Fourier series.
• In Fourier series complicated periodic functionsare written as the sum of simple wavesmathematically represented by sines and cosines.
• Due to the properties of sine and cosine it ispossible to recover the amount of each wave inth b i t l
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the sum by an integral.
• In many cases it is desirable to use Euler'sformula, which states that ei2πθ=cos2πθ+isin2πθ,to write Fourier series in terms of the basicwaves ei2πθ.
Fourier Transform
• From sines and cosines to complex exponentialsFrom sines and cosines to complex exponentialsmakes it necessary for the Fourier coefficientsto be complex valued. Complex number givesboth the amplitude (or size) of the wave presentin the function and the phase (or the initialangle) of the wave.
Th F i i l b d f i di
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• The Fourier series can only be used for periodicsignals.
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Fourier Transform
• How can the results be extended for Aperiodicsignals such as g(t) of limited length T ?
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First: Construct a new periodic signal gTo(t) formedby repeating the signal g(t) every T seconds
3.1 Aperiodic Signal Representation by Fourier Integral
by repeating the signal g(t) every To seconds.
• To is made long enough to avoid overlappingbetween the repeating pulses
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3.1 Aperiodic Signal Representation by Fourier Integral
Second: Calculate the exponential Fourier series ofgT (t) with ω =2П/TgTo(t) with ωo 2П/To
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Third: Let the period
3.1 Aperiodic Signal Representation by Fourier Integral
• Integrating gTo(t) over (-To/2,To/2) is the sameas integrating g(t) over (-∞ ∞) thereforeas integrating g(t) over ( , ), therefore
• Observe that the nature of the spectrumchanges as To increases. Let us define G(w); a
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g o ( )continuous function of ω
Then
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3.1 Aperiodic Signal Representation by Fourier Integral
• Fourier coefficients Dn are (1/To times) thesamples of G(ω) uniformly spaced at ωo rad/sec.Therefore (1/To) G(ω) is the envelope for thecoefficients Dn.
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3.1 Aperiodic Signal Representation by Fourier Integral
• Let To by doubling To repeatedly. Doubling Tohalves the fundamental frequency ωo, so thatthere are now twice as many samples in thespectrum.
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3.1 Aperiodic Signal Representation by Fourier Integral
• If we continue doubling To repeatedly, thespectrum becomes denser while its magnitudebecomes smaller, but the relative shape of theenvelope will remain the same.
T 0w0DSpectral components are spaced
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oT 0ow0nDp p p
at zero (infinitesimal) interval
Then Fourier series can be expressed as:
Fourier Transform
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Fourier Transform
gTo(t) can be expressed as a sum of everlastingexponentials of frequencies
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The sum on the right-hand side can be viewed asthe area under the function G(w)ejωt
The sum on the right-hand side can be viewed asthe area under the function G(w)ejωt
Fourier Transform
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Fourier Transform
G(w) is the direct Fourier transform of g(t)
( ) h F f f G( )g(t) is the inverse Fourier transform of G(w)
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Fourier Transform
G(w) is complex. To plot the spectrum G(w) as afunction of ω, we have both amplitude and phasespectra:
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Conjugate Symmetry Property
If g(t) is a real function of t
Th G( ) d G( ) lThen G(ω) and G(-ω) are complex conjugates:
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For real g(t),
The amplitude spectrum is an even function
The phase spectrum θg(ω) is an odd function of ω.
Example 3.1
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Example 3.1
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Example 3.1
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Linearity of the Fourier Transform
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3.2 Transforms of Some Useful Functions
A unit gate function rect(x) has a unit height andunit width centered at the origin.
Expanded by τ
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The denominator of the argument of rect(x/τ) is thewidth of the pulse.
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Unit Triangle FunctionA unit triangle function ∆(x) has a unit height anda unit width centered at the origin.
Expanded by τ
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The denominator of the argument of ∆(x/τ) is the widthof the pulse.
Interpolation Sinc Function
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L'Hôpital's Rule
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L'Hôpital's Rule
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Interpolation Sinc FunctionFor a sinc(3ω/7) the first zero occurs at ω= 7П/3.This is because the argument 3ω/7=П whenω=7П/3ω=7П/3.
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Example 3.2
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Example 3.2
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Bandwidth of rect(t/τ)• Most of the signal Energy of the spectrum of
the rect function is in the lower frequencycomponentscomponents.
• Signal Bandwidth: is the difference between thehighest (significant) frequency and the lowest(significant) frequency in the signal spectrum.
• Much of the spectrum of the rect function isconcentrated within the first lobe (from w=0 to
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concentrated within the first lobe (from w=0 tow=2П/τ.
• A rough estimate of the bandwidth of arectangular pulse of width τ is 2П/τ rad/sec or1/τ Hz.
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Bandwidth of rect(t/τ)• To compute the bandwidth, one must consider
the spectrum only for positive values of w.
• The trigonometric spectrum exists only forpositive frequencies.
• The negative frequencies occur because we useexponential spectra for mathematicalconvenience. Each sinusoid coswnt appears of twoexponential compnents with frequencies w and
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exponential compnents with frequencies wn andw-n.
• In reality there is only one component offrequency which is the wn.
Example 3.3
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Interpolation Sinc FunctionThe function [sin(x)]/x is denoted by sinc(x). It isalso known as the filtering or interpolating function.
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Example 3.4
or
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Example 3.5
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We also have:
Example 3.6
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Example 3.6
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Example 3.7
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Example 3.7The transform of sgn (t) can be obtained byconsidering the sgn as a sum of two exponentials (inthe limit a goes to zero)the limit a goes to zero).
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Some Properties of the Fourier Transform
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Time Frequency Duality
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Symmetry Property
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Example 3.8Apply the symmetry property to the pair that areshown below
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Example 3.8
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Example 3.8
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Example 3.8
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Scaling Property
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Scaling Property
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The scaling property states that the time compressionof a signal results in its spectral expansion, and timeexpansion of the signal results in its spectralcompression.
If g(t) is wider, its spectrum is narrower and vise versa.
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Example 3.9
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Example 3.9
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Time Shifting Property
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Time delay in a signal causes a linear phase shift in itsspectrum.
Example 3.10
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Frequency Shifting Property
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Multiplication of a signal by a factor of ejwot
shifts its spectrum by w=wo.
Scaling PropertyChanging wo to –wo yields
ejwot is not a real function that can be generated.
In practice frequency shift is achieved bymultiplying g(t) by a sinusoid as:
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Scaling Property• Multiplication of a sinusoid cos (wot) by g(t)
amounts to modulating the sinusoid amplitude.
• This type of modulation is called amplitudemodulation.
cos (wot) is called the carrier.
The signal g(t) is called the modulating signal.
g(t) cos (w t) is called the modulated signal
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g(t) cos (wot) is called the modulated signal.
To sketch g(t) cos (wot)
Amplitude Modulation
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Example 3.12
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The Fourier transform of g(t) is
Example 3.12
The Fourier transform of g(t) is
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Example 3.12
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Bandpass SignalsIf gc(t) and gs(t) are low-pass signals, each with abandwidth of B Hz or 2ПB rad/sec, then thesignals g (t) cos w t and g (t) sin w t are bothsignals gc(t) cos wot and gs(t) sin wot are bothbandpass signals occupying the same band, and eachhaving a bandwidth of 4ПB rad/sec.
A linear combination of these signals will also be abandpass signal occupying the same band as that ofeither signal and with the same bandwidth 4ПB
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grad/sec
A general bandpass signal can be expressed as
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Bandpass Signals
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Bandpass Signals
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Slowly varying envelope
Slowly varying phase
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Example 3.13
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Example 3.13
From Example 2.12
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Convolution
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Convolution
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Bandwidth of the Product of two Signals
• If g1(t) and g2(t) have bandwidths B1 and B2 Hz,respectively. Then the bandwidth of g1(t) g2(t) isB1 + B2 Hz.
• Consequently, the bandwidth of g(t) is B Hz,then the bandwidth of g2(t) is 2B Hz, and theb d id h f ( ) i B H
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bandwidth of gn(t) is nB Hz.
Example 3.14
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Example 3.14
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Example 3.14
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Time Differentiation and Time Integration
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Example 3.15
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Example 3.15
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Example 3.15
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Signal Transmission Through a Linear System
For a linear, time invariant, continuous-time systemthe input-output relationship is given by
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g(t) is the input, Y(t) is the output, and
h(t) is the unit impulse response
Signal Distortion During Transmission
The transmission of an input signal g(t) through asystem changes it into the output signal y(t).
G(w) and Y(w) are the spectra of the input and the
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G(w) and Y(w) are the spectra of the input and theoutput.
H(w) is the spectral response of the system.
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Signal Distortion During Transmission
Writing the equation in Polar form
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Signal Distortion During Transmission
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Distortionless Transmission• Transmission is said to be distortionless if the input and
the output have identical wave shapes within amultiplicative constant.mu p n n .
• A delayed output that retains the input waveform is alsoconsidered distortionless.
• Thus, in distortionless transmission, the input g(t) andthe output y(t) satisfy the condition
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Example 3.12
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Example 3.16
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Example 3.12
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Example 3.12
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Example 3.12
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Ideal and Practical Filters
The ideal low-pass filter allows all components beloww=W rad/s to pass without distortion andsuppresses all components above w=W.
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Ideal and Practical Filters
The ideal high-pass and bandpass filtercharacteristics are shown below.
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Signal Distortion During Transmission
The ideal low-pass filter has a linear phase of slope-td, which results in a time delay of td seconds forall its input components of frequencies below Wrad/s.
If the input is a signal g(t) band-limited to W rad/s,the output y(t) is g(t) delayed by td, that is,
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For this filter |H(w)| = rect (w/2W), and θh(w)=-wtd, so that
Ideal and Practical FiltersThe unit impulse response h(t) of this filter is:
The impulse response h(t) is not realizable. Onepractical approach to filter design is to cut off thetail of h(t) for t<0 The resulting causal impulse
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tail of h(t) for t<0. The resulting causal impulseresponse h(t)
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Non-causal Causal
Ideal and Practical Filters
If td is sufficiently large, h(t) will be a closeapproximation of h(t), and the resulting filter ii (w)
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pp ( ), g ( )will be a good approximation of an ideal filter.
Theoretically a delay td = ∞ is needed to realizethe ideal characteristics. But a delay td of threeor four times П/W will make h(t) a reasonably closeversion of h(t-td).
• The truncation operation [cutting the tail of h(t)to make it causal], however, creates some
Ideal and Practical Filters
unsuspected problems of spectral spread andleakage
• This can be partly corrected by truncating h(t)gradually (rather than abruptly) using a taperedwindow function.
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• Practical (realizable) filter characteristics aregradual, without jump discontinuities in theamplitude response |H(w)|.
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The well-known Butterworth filters, for example,have amplitude response
Butterworth Filters
The amplitude responseapproaches an ideallow-pass behavior as
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n—›∞.
the Butterworth filter(half-power -3dB or aratio of 1/√2=0.707)bandwidth is B Hz. Alsocalled cutoff frequency.
Comparison of Butterworth Filter (n=4) with an ideal Filter
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As n—›∞, the amplitude response approaches ideal, but thecorresponding phase response is badly distorted in the vicinityof the cutoff frequency B Hz. A certain trade-off existsbetween ideal magnitude and ideal phase characteristics.
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A signal transmitted over a channel is distortedbecause: of various channel imperfections.
3.6 Signal Distortion over a Communication Channel
because: of various channel imperfections.
1. Linear Distortion
Signal distortion can be caused over a linear time-invariant channel by nonideal characteristics ofeither the magnitude, the phase, or both.
If l (t) i t itt d th h h
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If a pulse g(t) is transmitted through such achannel. Spreading, or dispersion, of the pulse willoccur if either the amplitude response or the phaseresponse, or both, are nonideal.
Linear Distortion
A distortionless channel multiplies each componentb h f d d l h bby the same factor and delays each component bythe same amount of time.
If the amplitude response of the channel is notideal [that is, |H(w)| is not equal to a constant],then the pulse will spread out (see the followingexample)
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example).
The same thing happens if the channel phasecharacteristic is not ideal [θh (w) -wtd].
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Dispersion of the pulse is undesirable in a TDM
Linear Distortion
system, because pulse spreading causes interferencewith a neighboring pulse and consequently with aneighboring channel (crosstalk).
For an FDM system, this type of distortion causesdistortion (dispersion) in each multiplexed signal, butno interference occurs with a neighboring channel
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no interference occurs with a neighboring channel.This is because in FDM, each of the multiplexedsignals occupies a band not occupied by any othersignal.
Example 3.17
A low-pass filter transfer function H(w) is given by
A pulse g(t) band-limited to B Hz is applied at theinput of this filter. Find the output y(t).
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This filter has ideal phase and nonideal magnitudecharacteristics.
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The output is actually g(t) + (k/2)[g(t-T)+g(t+T)]delayed by td. It consists of g(t) and its echoesshifted by ±td. The dispersion of the pulse causedby its echoes is evident from the figure below.
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We shall consider a simple case of a memorilessnonlinear channel where the input g and the output y
2. Distortion Caused by Channel Nonlinearities
are related by some nonlinear equation,
The right-hand side of this equation can beexpanded in a McLaurin's series as
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If the bandwidth of g(t) is B Hz, then thebandwidth of gk(t) is kB Hz. Then, the bandwidth ofy(t) is kB Hz
Distortion Caused by Channel Nonlinearities
y(t) is kB Hz.
The output spectrum spreads well beyond the inputspectrum, and the output signal contains newfrequency components not contained in the inputsignal.
If a signal is transmitted over a nonlinear channel
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If a signal is transmitted over a nonlinear channel,the nonlinearity not only distorts the signal, butalso causes interference with other signals on thechannel because of its spectral dispersion(spreading) which will cause a serious interferenceproblem in FDM systems (but not in TDM systems).
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Example 3.18
The input x(t) and the output y(t) of a certainnonlinear channel are related as
1. Find the output signal y(t) and its spectrum Y(w)if the input signal is x(t)=(1000/π) sinc(1000t).
2. Verify that the bandwidth of the output signal istwice that of the input signal This is the result
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twice that of the input signal. This is the resultof signal squaring.
3. Can the signal x(t) be recovered (withoutdistortion) from the output y(t)?
Example 3.18
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Example 3.18
Observe that 0.316 sinc2(1000t) is the unwanted(distortion) term in the received signal.
Input signal t X( )
Spectrum of the di t ti t
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spectrum X(w) distortion term
Received signal spectrum Y(w)
Example 3.18
We make the following observations:
1 The bandwidth of the received signal y(t) is twice1. The bandwidth of the received signal y(t) is twicethat of the input signal x (t) (because of signalsquaring).
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Input signal spectrum X(w)
Received signal spectrum Y(w)
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Example 3.18
2. The received signal contains the input signal x(t)plus an unwanted signal (1000/π) sinc2(1000t).Note that the desired signal and the distortionsignal spectra overlap, and it is impossible torecover the signal x(t) from the received signaly(t) without some distortion.
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Received signal spectrum Y(w)
Example 3.18
3. We can reduce the distortion by passing thereceived signal through a low-pass filter ofbandwidth I000 rad/s. Observe that the outputof this filter is the desired input signal x(t) withsome residual distortion.
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Received signal spectrum Y(w)
Spectrum of the Received signal after the low-pass filtering.
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Example 3.18
4. We have an additional problem of interferencewith other signals if the input signal x (t) isfrequency-division multiplexed along with severalother signals on this channel. This means thatseveral signals occupying nonoverlapping frequencybands are transmitted simultaneously on thesame channel.
Spreading of the spectrum X(w) outside its original
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Spreading of the spectrum X(w) outside its originalband of 1000 rad/s will interfere with the signal inthe band of 1000 to 2000 rad/s. Thus, in additionto the distortion of x(t), we also have aninterference with the neighboring band.
Example 3.18
5. If x(t) were a digital signal consisting of a pulsetrain, each pulse would be distorted, but therewould be no interference with the neighboringpulses. Moreover even with distorted pulses, datacan be received without loss because digitalcommunication can withstand considerable pulsedistortion without loss of information.
Thus if this channel were used to transmit a TDM
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Thus, if this channel were used to transmit a TDMsignal consisting of two interleaved pulse trains, thedata in the two trains would be recovered at thereceiver.
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Example 3.18
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3. Distortion Caused by Multipath Effects
A multipath transmission takes place when atransmitted signal arrives at the receiver by two ormore paths of different delays.
In radio links, the signal can be received by directpath between the transmitting and the receivingantennas and also by reflections from other objects,such as hills, buildings, and so on.
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In this case the transmission channel can berepresented as several channels in parallel, each witha different relative attenuation and a different timedelay.
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Distortion Caused by Multipath Effects
For multipath signal propagation
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Distortion Caused by Multipath Effects
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Distortion Caused by Multipath Effects
Let us consider the case of only two paths: onewith a unity gain and a delay td, and the other witha gain α and a delay td+∆t, as shown below.
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Distortion Caused by Multipath Effects
The transfer functions of the two paths are given bye-jwtd and αe-jw(td+∆t), respectively. The overalltransfer function of such a channel is H(w), given by
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Both the magnitude and the phase characteristicsof H(w) are periodic in ω with a period of 2π/∆t.
Distortion Caused by Multipath Effects
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The multipath transmission, therefore, causesnonidealities in the magnitude and the phase
Distortion Caused by Multipath Effects
nonidealities in the magnitude and the phasecharacteristics of the channel and will cause lineardistortion (pulse dispersion).
If the gains of the two paths are very close, thatis, α≈1, the signals received by the two paths canvery nearly cancel each other at certain frequencies,
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y y qwhere their phases are π rad apart (destructiveinterference).
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At frequencies where ω=nπ/∆t (n odd), cos ω∆t=-1,
Distortion Caused by Multipath Effects
q ( ), ,and lH(w)l≈0 when α≈1. These frequencies are themultipath null frequencies.
At frequencies ω=nπ/∆t (n even), the two signalsinterfere constructively to enhance the gain. Suchchannels cause frequency-selective fading oftransmitted signals Such distortion can be partly
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transmitted signals. Such distortion can be partlycorrected by using the tapped delay-line equalizer.
Thus far, the channel characteristics were assumed
4. Fading Channels
,to be constant with time. In practice, we encounterchannels whose transmission characteristics vary withtime. These include troposcatter channels andchannels using the ionosphere for radio reflection toachieve longdistance communication.
The time variations of the channel properties arise
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The time variations of the channel properties arisebecause of semiperiodic and random changes in thepropagation characteristics of the medium.
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The reflection properties of the ionosphere, forexample are related to meteorological conditions
Fading Channels
example, are related to meteorological conditionsthat change seasonally, daily, and even from hourto hour, much the same way as does the weather.Periods of sudden storms also occur. Hence, theeffective channel transfer function variessemiperiodically and randomly, causing randomattenuation of the signal This phenomenon is known
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attenuation of the signal. This phenomenon is knownas fading. One way to reduce the effects of fadingis to use automatic gain control (AGC).
Fading may be strongly frequency dependent where
Fading Channels
Fading may be strongly frequency dependent wheredifferent frequency components are affectedunequally. Such fading is known as frequency-selective fading and can cause serious problems incommunication. Multipath propagation can causefrequency-selective fading.
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The energy Eg of a signal g(t) is defined as the areaunder lg(t)l2 • We can also determine the signal
Signal Energy and Energy Spectral Density
energy from its Fourier transform G(w) throughParseval's theorem.
Signal energy can be related to the signal spectrumG(w) by:
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Verify Parseval’s theorem for the signal g(t)=e-atu(t)(a>0).
Example 3.19
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The energy spectral density (ESD) Ψg(t) is given by:
Signal Energy and Energy Spectral Density
Th ESD f th i l (t) at (t) i :
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The ESD of the signal g(t)=e-atu(t) is:
Most of the signal energy is contained within acertain band of B Hz. Therefore, we can supress
Essential Bandwidth of a Signal
the signal spectrum beyond B Hz with little effecton the signal shape and energy.
The bandwidth B is called the essentail bandwidth ofthe signal.
Supression of all the spectral components of g(t)
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beyond the essential bandwidth results in a signalg^(t).
If we use 95% criterion for the essential bandwidth,the energy of the error g(t)-g^(t) is 5% of Eg.
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Estimate the essential bandwidth W rad/sec of thesignal g(t)=e-atu(t) if the essential band is required
Example 3.20
to contain 95% of the signal energy.
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Example 3.20
The signal Eg is 1/2π times the area under the ESD,which is found to be 1/2a.
Let W rad/sec be the essential bandwidth, whichcontains 95% of the total signal energy Eg. This
1/2 ti th h d d
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means 1/2π times the shaded area;
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Example 3.20
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This means that all the remaining spectralcomponents in the band from 12.706 to ∞ contributeonly 5% of the signal energy.
Estimate the essential bandwidth of the arectangular pulse g(t)=rect(t/T), where the essential
Example 3.21
bandwidth is to contain at least 90% of the pulseenergy.
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Example 3.21
The energy EW within the band from 0 to W rad/sec
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gy Wis given by:
Setting wT=x in this integral so that dw=(1/T) dx;
Example 3.21
Eg=T, we have;
Note that 90.28% of
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the total energy of thepulse g(t) is containedwithin the bandW=2π/T rad/sec orB=1/T Hz.
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Let g(t) be a baseband signal band-limited to b-Hz.The amplitude modulation ψ(t) is:
Energy of Modulated Signals
The ESD of the modulated signal is
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Energy of Modulated Signals
The modulation shiftsthe ESD of g(t) by ±ωo.
The area under ФΨ(w) ishalf the area underΨg(w)
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For a real signal g(t), the autocorrelation functionΨg(τ) is given by:
Time Autocorrelation Function and the Energy Spectral Density
Setting x=t+τ
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In this equation x is a dummy variable and could bereplaced by t.
This shows that for a real g(t), the autocorrelationfunction is an even function of τ
Time Autocorrelation Function and the Energy Spectral Density
We now show that the ESD Ψg(ω)=lG(ω)l2 is theFourier transform of the autocorrelation functionψg(τ)
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The autocorrelation ψg(τ) is the convolution of g(τ)with g(-τ)
Time Autocorrelation Function and the Energy Spectral Density
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Find the time autocorrelation function of the signalg(t)=e-atu(t), and from it determine the ESD of g(t).
Example 3.22
This is valid for positive τ. We can perform a similar procedure fornegative τ. However, for real g(t), ψg(τ) is an even function of τ.
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g , g( ), ψg( )
ESD of Ψg(ω) is the Fourier transform of ψg(τ)
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Example 3.22
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ESD of the Input and the Output
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3.8 Signals Power and Power Spectral Density
Defining a truncated signal gT(t) as
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If the signal g(t) is a power signal, then its power isfinite, and the truncated signal gT(t) is an energy
Power Spectral Density (PSD)
signal as long as T is finite. From Parseval’s Theorem
Pg, the power of g(t), is given by
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The power spectral density (PSD) Sg(w) is defined as:
Power Spectral Density (PSD)
The power spectral density (PSD) Sg(w) is defined as:
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The time autocorrelation function Rg(τ) of a realpower signal g(t) is defined as:
Time Autocorrelation Function of Power Signals
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The PSD Sg(w) is the Fourier transform of Rg(τ)
Time Autocorrelation Function of Power Signals
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The concept and relationships for signal power areparallel to those for signal energy.
Time Autocorrelation Function of Power Signals
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The signal power is the time average or mean of itssquared value. In other words Pg is the mean square
Signal Power is its Mean Square Value
gof g(t).
A wavy overline is used to denote a time average.
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The PSD Sg(w) represents the power per unitbandwidth (in hertz) of the spectral components at
Interpretation of Power Spectral Density
the frequency w. The power contributed by thespectral components within the band w1 to w2 is givenby:
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The Fourier transform is available only fordeterministic signals, which can be described as
Autocorrelation Method: a Powerful Tool
functions of time.
The random message signals that occur incommunication problems, can not be described asfunctions of time, and its impossible to find theirFourier transforms.
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The autocorrelation function for such signals can bedetermined from their statistical information. Thisallows us to determine the PSD (the spectralinformation) of such a signal.
A random binary pulse train g(t) is shown below. Thepulse width is Tb/2, and one binary digit is
Example 3.23
transmitted every Tb seconds. A binary 1 istransmitted by the positive pulse, and a binary 0 istransmitted by the negative pulse. The two symbolsare equally likely and occur randomly. We shalldetermine the autocorrelation function, the PSD, andthe essential bandwidth of this signal.
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We cannot describe this signal as a function of timebecause the precise waveform is not known due to
Example 3.23
its random nature.
We do know its behavior in terms of the averages(the statistical information).
The autocorrelation function, being an averageparameter (time average) of the signal, is
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determinable from the given statistical (average)information.
The figure below shows g(t) by solid lines and g(t-τ)by dashed lines.
Example 3.23
To determine the integrand on the right-hand sideof the autocorrelation equation, we multiply g(t) withg(t-τ), find the area under the product g(t)g(t-τ),and divide it by the averaging interval T. Let therebe N bits (pulses) during this interval T so that T= NTb and as T —›∞ N —›∞
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= NTb, and as T ›∞, N ›∞.
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Let us first consider the case of τ<Tb/2. In thiscase there is an overlap (shown by the shaded
i ) b t h l f (t) d th t f (t
Example 3.23
region) between each pulse of g(t) and that of g(t-τ). The area under the product g(t)g(t-τ) is Tb/2-τfor each pulse. Since there are N pulses during theaveraging interval, then the total area under g(t)g(t-τ) is N(Tb/2-τ),
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Because Rg(τ) is an even function of τ,
Example 3.23
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As we increase τ beyond Tb/2, there will beoverlap between each pulse and its immediate
Example 3.23
overlap between each pulse and its immediateneighbor. The two overlapping pulses are equallylikely to be of the same polarity or of oppositepolarity. Their product is equally likely to be 1 or-1 over the overlapping interval.
On the average, half the pulse products will be 1( iti iti ti ti l
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(positive-positive or negative-negative pulsecombinations), and the remaining half pulse productswill be -1 (positive-negative or negative-positivecombinations).
Consequently, the area under g(t)g(t- τ) will be zerowhen averaged over an infinitely large time (T —›∞),
Example 3.23
The autocorrelation function in this case is thetriangle function 0.5∆(t-Tb). The PSD is the Fouriertransform of 0.5∆(t-Tb), which is found in Example3.15 as
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3.15 as
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The PSD is the square of the sinc function shownbelow. From the result in Example 3.21, we concludethat the 90 28% of the area of this spectrum is
Example 3.23
that the 90.28% of the area of this spectrum iscontained within the band from 0 to 4П/Tb rad/s, orfrom 0 to 2/Tb Hz. Thus, the essential bandwidthmay be taken as 2/Tb Hz (assuming a 90% powercriterion).
This example illustrates dramatically how the
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p yautocorrelation function can be used to obtain thespectral information of a (random) signal whereconventional means of obtaining the Fourier spectrumare not usable.
Because the PSD is a time average of ESDs, the
Input and Output Power Spectral Densities
relationship between the input and output signal PSDsof a linear time-invariant (LTI) system is similar tothat of ESDs.
If g(t) and y(t) are the input and output signals ofan LTI system with transfer function H(w), then
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A noise signal ni(t) with PSD Sni(w)=K is applied atthe input of an ideal differentiator. Determine the
Example 3.24
PSD and the power of the output noise signal n0 (t).
The transfer function of an ideal differentiator is( ) f h h d d l
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H(w)=jw. If the noise at the demodulator output isn0(t),
The output PSD Sno(w) is parabolic, as shown below.The output noise power N0 is l/2П times the area
Example 3.24
under the output PSD. Therefore,
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For a power signal g(t), if
PSD of Modulated Signals
Then the PSD SΦ (w) of the modulated signal Φ(t) isgiven by
Thus modulation shifts the PSD of g(t) by ±w • The
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Thus, modulation shifts the PSD of g(t) by ±w0• Thepower of Φ(t) is half the power of g(t), that is,
Homework #3 •Solve the following problems: Due to one weekfrom today
•3.1-8
•3.3-5
•3.3-6
•3.3-7
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•3.3-10
•3.7-5
•3.8-2
