1 Object:mathematical tools to describe and analyze the signals Fourier series and transform Important function:Dirac delta function,rectangular function, periodic function and sinc function and their Fourier transforms frequency analyze (time function and his spectrum) some properties of signal (DC value ,root mean square value,…) power spectral density and autocorrelation function linear systems:linear time-invariant systems,impulse response,transfer function,distortionless transmission bandwidth concept :baseband,passband and bandlimited signals and noise *sampling theorem ( dimensionality theorem) summary Chap.2 Signals and Spectra
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1 Object: mathematical tools to describe and analyze the signals Fourier series and transform Important function:Dirac delta function,rectangular function,
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Object:mathematical tools to describe and analyze the signals
Fourier series and transform
Important function:Dirac delta function,rectangular function, periodic function and sinc function and their Fourier transforms
frequency analyze (time function and his spectrum)
some properties of signal (DC value ,root mean square value,…)
power spectral density and autocorrelation function
linear systems:linear time-invariant systems,impulse response,transfer function,distortionless transmission
bandwidth concept:baseband,passband and bandlimited signals and noise
*sampling theorem (dimensionality theorem)
summary
Chap.2 Signals and Spectra
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• Signal:desired part of waveforms; Noise:undesired part
• Electric signal’s form:voltage v(t) or current i(t) (time function)
• In this chapter,all signals are deterministic.• But in communication systems,we will be face the
stochastic waveforms
Deterministic results stochastic results by analogy
Signal analysis:first importance
2-1. Properties of Signals and Noise
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• Non zero values over a finite time interval
• non zero values over a finite frequency interval
• a continuous time function
• a finite peak value
• only real values
In general,the waveform is denoted by w(t)
When t→±∞,we have w(t) →0,but w(t) is defined over (+∞,-∞)
The math model of waveform can violate some or all above conditions.
Ex. w(t)=sinωt,physically this waveform can not be existed.
Physically realizable waveforms
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Waveforms:
• signal or noise
• digital or analog
• deterministic or nondeterministic(stochastic)
• physically realizable or nonphysically realizable
• power type or energy type
• periodic or nonperiodic
Power type:the average power of the waveform is finite(math model)
Energy type:the average energy of the waveform is finite(all physically realizable signal)
The classifications of waveforms
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• Time average operator:dc(direct current) value of time function
Definition: the time average operation is given by:
〈 [·] 〉 =lim1/T-T/2∫
T/2[·]dt
〈 [·] 〉 is time average operator. The operator is linear.(Why?)
Definition : w(t) is periodic with period T0 if
w(t)=w(t+ T0) for all t
where T0 is smallest positive number that satisfies above relationship.
Theorem:if w(t) is periodic,the time average operation can be reduced to 〈 [·] 〉 =1/T-T/2-a∫
T/2+a[·]dt
where T is period of w(t)
Some important math operations
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• Definition:the dc value of w(t) is given by its time average,
〈 w(t) 〉 .
Wdc= 〈 w(t) 〉 =lim1/T-T/2∫
T/2w(t)dt
or
Wdc= 〈 w(t) 〉 =1/(t2-t1)∫ w(t)dt
Power• Definition:the instantaneous power is given by:
p(t) = v(t)i(t)
and the average power is : P=<p(t)>=<v(t)i(t)>
DC value
+
v(t)
-
i(t)
circuit
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• Definition:the root mean square (rms) value of w(t)is given by: Wrms=[<w2(t)>]1/2
• Theorem:if a load (R) is resistive,the average power is:
P= <v2(t)>/R= <i2(t)>R= V2rms/R=I 2
rmsR
• Definition:if R=1Ω,the average power is called normalized power.
Then i(t) = v(t) = w(t) and P= <w2(t)>
Energy and Power Waveforms:
• Definition:w(t) is a power waveform if and only if the normalized power P is finite and nonzero(0<P<∞).
• Definition:the total normalized energy is given by:
Rms Value and Normalized Power
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• Definition:w(t) is an energy waveform if and only if the total normalized energy is finite and nonzero (0<E<∞).
Waveform: power signal or energy signal
Energy finite Average Power=0
Power finite Energy=∞
Physically realizable waveform:Energy waveform
Periodic waveform:Power waveform
Decibel
• Definition:the decibel gain of a circuit is given by
dB=10log(average power out/average power in) =10log(Pout/Pin)
2/
2
2 )(1
lim
T
TT
dttwT
E
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For normalized power case(R=1Ω),we have:
dB=20log(Vrms out /Vrms in)= 20log(Irms out /Irms in)
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w(t),voltage or current,time function analysis in time domain. Their fluctuation as a function of time is an important characteristic to analyze the signal’s comportment when they present in the transmission channel or other communication’s units. Frequency analysis of signal. Tool to realize the frequency domain analysis of signal Fourier Transformation
• Definition:The Fourier Transform (FT) of w(t) is :
W(f)=F[w(t)]= -∞∫∞w(t)exp[-j2πft]dt
f :frequency (unit:Hz if t is in sec)
In general,W(f) is called a two-sided spectrum of w(t)
Some properties: W(f) is a complex function
so W(f)=X(f)+jY(f)=│W(f)│exp[jθ(f)]
Fourier Transform and Spectra
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Here we have the polar (or magnitude-phase) form of FT: │W(f)│=[X2(f)+Y2(f)]1/2:magnitude spectrum
θ(f)=arctg[Y(f)/X(f)]:phase spectrum
Inverse Fourier transform:
w(t)=F-1[W(f)]= -∞∫∞W(f)exp[j2πft]df
Ex. Spectrum of an exponential pulse:
w(t)=e-t, t>o
W(f)=0∫∞ e-t exp[j2πft]dt=1/(1+j2πf)
X(f)=
Y(f)=
│W(f)│=[X2(f)+Y2(f)]1/2, θ(f)=arctg[Y(f)/X(f)]
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• Theorem:Spectral symmetry of real signals.
W(t) is real, then W(-f)=W*(f)
Proof. See text
Deduction: │W(-f)│=│W(f)│:The magnitude spectrum is even function of f
θ(-f)= - θ(f): the phase spectrum is odd
Summary:
• f,frequency (Hz),an FT’s parameter that specifies w(t)’s interested frequency.
• FT looks for frequency f in w(t) over all time
• W(f) is complex in general
• w(t) real,then W(-f)=W*(f)
Properties of Fourier Transforms
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• Parseval’s theorem:
-∞∫∞w1(t) w2*(t) dt=-∞∫∞ W1(f) W2
*(f)df
if w1(t) =w2(t) =w(t),we have
E= -∞∫∞w2(t)dt=-∞∫∞ │W(f)│2df
Proof: directly from FT
Energy spectral density(ESD)• Definition:The ESD is defined for energy waveforms by:
E(f)= │W(f)│2 J/Hz
By using Parseval’s theorem we have
E =-∞∫∞ E(f)df
Parseval’s theorem
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• For power waveforms,we have a similar function called PSD.(see later)
Another properties of FT:
• W(f) is real if w(t) is even
• W(f) is complex if w(t) is odd
We have some basic and important FT’s theorems at 附录 A.
Most important theorems:
time delay :w(t-Td) W(f)e-j ωTd
frequency translation:
w(t)cos(ωct+θ) 1/2[ W(f-fc)ejθ+W(f+ fc)e-jθ]
Power spectral density(PSD)
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Convolution:w1(t)*w2(t) W1(f)W2(f)
Differentiator: dw(t)/dt j2πfW(f)
Integrator: -∞∫tw(t)dt W(f)/ (j2πf)+1/2W(0)δ(f)
Frequency translation:(w(t) is real)
we can use the FT’s definition to prove these theorem.(Home works)
W(f)
f
f
F[w(t)cosωct]
fc
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• Dirac delta function is very useful (perhaps the most useful)in communication system’s analysis.
• Definition: δ(x) is defined by
-∞∫∞w(x)δ(x) dx=w(0)
where w(x) is any function that is continuous at x=0.
Or we can equally define the δ(x) as:
-∞∫∞δ(x) dx=1
and
δ(x)=∞ when x=0
δ(x)=0 when x≠0
So we can use two delta functions’ definitions without difference.
Dirac delta function and unit step function
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• The sifting property:
-∞∫∞w(x)δ(x-x0) dx=w(x0)
• An useful delta function’s expression:
δ(x) =-∞∫∞e±j2πxy dy
Proof: we have delta function’s FT:
-∞∫∞e-j2πft δ(t) dt=e0=1
and take the inverse Fourier transform of above equation, then
δ(t) =-∞∫∞e+j2πft df
• δ(x) is even: δ(x) = δ(-x)
Delta function’s properties
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• Unit step function:closely related with δ(x)
• Definition: u(t) is defined by:
u(t)=1 for t>0 and u(t)=0 for t<0
Properties: -∞∫x δ(x) dx=u(x)
and
du(t)/dt= δ(t)
Ex. Spectrum of a sinusoid
v(t)=Asinω0t, ω0=2πf0
from FT ,we have:
V(f) = -∞∫∞(A/2j)(ej2πf0t - e-j2πf0t )e-j2πft dt
= j(A/2)[δ(f+f0) -δ(f-f0)]
│V(f)│=(A/2) δ(f-f0)+ (A/2) δ(f+f0),
θ(f)=π/2 for f>0 and θ(f)= -π/2 for f<0
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│V(f)│
ff0- f0
A/2
f
θ(f)
Magnitude spectrumPhase spectrum
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• Conclusion:A sinusoid waveform has mathematically two frequency components( at f=±f0) and his magnitude spectrum is a line spectra.
Rectangular pulse:
• Definition:The rectangular pulse Π(·) is defined by:
Π(t/T)=1 for │t│≤T/2
Π(t/T)=0 for │t│≥T/2
• Definition:The function sinc(·) is defined by:
sinc(x)=(sinπx)/(πx)
or the function Sa (·) is difined by
Sa (x)=sinc(x/π)=sin x/x
Two very important functions in digital communication system’s analysis
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• Ex. Spectrum of a rectangular pulse
W(f)= -∞∫∞ Π(t/T)e-j2πft dt =-T/2∫ T/21e-j2πft dt
=Tsin(2πfT/2)/(2πfT/2)=TSa(πfT)
so we have: Π(t/T) TSa(πfT)
Time domain Frequency domain
Π(t/T)
tT/2-T/2
1
f
TSa(πfT)
T
1/T 2/T
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• For an ideal low pass filter, we have its time response:
Π(f/2W) 2WSa(2πWt)
• Conclusion:ideal LPF physically unrealizable
• The equivalent LPF plays a special role in digital comm.
• For triangular pulse,we have :
Λ(t/T) TSa2(πTf)
Π(f/2W)
W-W
1
f
2WSa(2πWt)
t
2W
1/2W-1/2W
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• Definition:the convolution of waveforms w1(t) and w2(t) gives a third function w3(t) defined by:
• Definition:A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin and negligible elsewhere.
• Definition:A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency f= ±fc ,where fc>>0.The spectral magnitude is negligible elsewhere. fc is called the carrier frequency.
• fc may be arbitrarily assigned.
• Definition:Modulation is the process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude and/or phase perturbation.This bandpass signal is called the modulated signal s(t),and the baseband source signal is called the modulating signal m(t).
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• All banpass waveforms can be represented by their complex envelope forms.
• Theorem:Any physical banpass waveform can be represented by:
v(t)=Re{g(t)ejωct}
Re{.}:real part of {.}.g(t) is called the complex envelope of v(t),and fc is the associated carrier frequency.Two other equivalent representations are:
v(t)=R(t)cos[ωct+θ(t)]
and
v(t)=x(t)cos ωct-y(t)sin ωct
where g(t)=x(t)+jy(t)=R(t) ejθ(t)
Complex envelope representation
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• Representation of modulated signals
• The modulated signals a special type of bandpass waveform
• So we have
s(t)=Re{g(t)ejωct}
the complex envelope is function of the modulating signal m(t): g(t)=g[m(t)]
g[.]: mapping function
All type of modulations can be represented by a special mapping function g[.].
• Theorem:If a bandpass waveform is represented by:
v(t)=Re{g(t)ejωct}
then the spectrum of the bandpass waveform is
V(f)=1/2[G(f-fc)+G*(-f-fc)]
and the PSD of the waveform is
Pv(f)=1/4[Pg(f-fc)+Pg(-f-fc)]
where G(f)=F[g(t)], Pg(f) is the PSD of g(t).
Proof:page 230
Spectrum of bandpass signals
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• One of most useful and important theorems in signal processing theory
• Mathematically speaking,the sampling theorem is an application of an orthogonal series expansion.
• Sampling theorem:any physical waveform may be represented over the interval -∞ < t < +∞ by:
w(t)=∑ansin{πfs[t-(n/fs)]}/{πfs[t-(n/fs)]}
where an=fs -∞∫∞ w(t) sin{πfs[t-(n/fs)]}/{πfs[t-(n/fs)]} dt
fs>0 (some convenient value)
Furthermore,if w(t) is bandlimited to B Hz and fs≥2B then w(t) can be represented by sampling function with
an=w(n/ fs)
sampling theorem
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• Proof: if we can show that
φn(t)=sin{πfs[t-(n/fs)]}/{πfs[t-(n/fs)]}
form a set of the orthogonal functions,then all waveform w(t) can be represented by {an}.
Orthogonal -∞∫∞ φn(t) φm*(t)dt =Knδnm
(see p.87)
The minimum sampling rate allowed to reconstruct a bandlimited waveform without error :(fs)min=2B
(fs)min is called the Nyquist frequency.
Commentary:using N sampling values to reproduce approximately a bandlimited waveform over an T0 interval
w(t)≈∑an φn(t) (n from nl to nl+N) and there N=fs T0≥2B T0
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• Impulse sampling
• According to sampling theorem, we can transmit only a set of discrete values w(n/fs) (called sampling values of w(t)) over a communication system or save these values in computer memory,and then using {w(n/fs) } with an appropriated method we can reconstruct w(t) without any error.
• The impulse train {w(n/fs) } can be presented by
ws(t)=∑ w(n/fs)δ(t-nTs)=w(t) ∑δ(t-nTs)
t
w(t)ws(t)
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• Waveforms and their spectrums
Let us examine the spectrum of w(t)’s sampling function:
ws(t)=∑ w(n/fs)δ(t-nTs)=w(t) ∑δ(t-nTs)
take FT on above equation,we have:
Ws(f)=∑w(n/fs)e-j2nπfTs=W(f)*[ ∑ e-j2nπfTs]
∑ e-j2nπfTs =? [∑ e-j2nπfTs]=∑δ(f-nfs)
so Ws(f)=1/Ts∑W(f-nfs)
w(t)
B-B
W(f)
f
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• The spectrum of ws(t) is:
• The spectrum of the impulse sampled waveform has different parts: a) unsampled waveform w(t)’s spectrum and b) the spectrum W(f) shifted in frequency every fs.
• We can conclude: if fs≥2B, the replicated spectra do not overlap and the original spectrum can be regenerated by chopping Ws(f) off above fs/2.
B-Bf
Ws(f)
fs/2fs/2
LPF
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• If fs≤2B ,then:
so we lost the some information of w(t).
Important remark:the sampling rate must be equal to 2B at least.If not,the sampling values can not represent exactly the bandlimited waveform.The recovered w(t) will be distorted because of the aliasing.
Ws(f)
fBfs
overlap
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• An explication of sampling theorem: dimensionality theorem
• Theorem: a real waveform may be completely specified by
N=2BT0
independent pieces of information that will describe the waveform over a T0 interval.N is said to be the number of dimensions required to specify the waveform and B is the absolute bandwidth of the waveform.
Important:The dimensionality theorem show that the information that can be conveyed by a bandlimited waveform or a bandlimited communication system is proportional to the product of the bandwidth of the system and the time allowed for transmission of the information.
Dimensionality theorem
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• The dimensionality theorem has profound implications in the design and performance of all type of communication systems.
• Ex. Radar system:the time-bandwidth product of received signal to be large superior performance
• Two way to be explained or applied the The dimensionality theorem :
• a) for a bandlimited waveform over a T0’s interval,how much values must be stored?
• b) to estimate the waveform’s bandwidth.
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• Spectral width of signal and noise is a very (or the most) important concept in communication systems.
• What is bandwidth?
• In engineering definitions,the bandwidth is taken to be the width of a positive frequency band.
• 5. Bounded spectrum bandwidth (out the band, PSD less 50dB below PSDmax)
• 6. Power bandwidth (99% power)
• 7. FCC bandwidth
Bandwidth of signals
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• Example (page 105):bandwidth for a BPSK signal
• Summary :
• Basic concepts:signal and noise (deterministic and stochastic), time domain analysis (dc value,rms value... )and frequency domain analysis (FT, spectra,linear system...),Fourier series , periodic function’s spectra (line spectra and continuous spectra ),bandlimited waveform,sampling theorem and its physical meaning,dimensionality theorem,bandwidth definitions