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

2

• 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

3

• 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

4

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

5

• 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

6

• 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

7

• 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

8

• 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

9

For normalized power case(R=1Ω),we have:

dB=20log(Vrms out /Vrms in)= 20log(Irms out /Irms in)

10

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

11

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)]

12

• 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

13

• 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

14

• 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)

15

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

16

• 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

17

• 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

18

• 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

19

│V(f)│

ff0- f0

A/2

f

θ(f)

Magnitude spectrumPhase spectrum

20

• 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

21

• 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

22

• 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

23

• Definition:the convolution of waveforms w1(t) and w2(t) gives a third function w3(t) defined by:

w3(t)= w1(t)*w2(t)= -∞∫∞ w1(λ)w2(t-λ) dλ

Ex.We have: Λ(t/T) = Π(t/T) * Π(t/T)

then F[Λ(t/T) ]=F[Π(t/T)]F[Π(t/T)] = TSa2(πTf)

Ex.Spectrum of a switched sinusoid:

w(t)= Π(t/T) Asin(ω0t)= Π(t/T) Acos(ω0t-π/2)

we have :Π(t/T) TSa(πfT)

Asin(ω0t) j(A/2)[δ(f+f0) -δ(f-f0)]

So W(f)= [TSa(πfT)] *{j(A/2)[δ(f+f0) -δ(f-f0)] }

= j(A/2){Sa[πT(f+f0)] - Sa[πT(f-f0)]}

Convolution

24

Time domain:

Frequency domain

W(t)

tT/2-T/2

│W(f)│

AT/2

f0 -f0

f

25

• Power spectral density

w(t) and its truncated waveform wT(t) :

wT(t)=w(t) -T/2<t<T/2 and wT(t) =0 t elsewhere

wT(t)’s average normalized power P:

P=lim 1/T -T/2∫ T/2wT

2(t) dt = lim 1/T -∞∫∞ wT2(t) dt

By Parseval’s theorem, we have

P=lim1/T -T/2∫ T/2│WT(f)│2df =-∞∫∞ lim[│WT(f)│2/T] df

where WT(f) is the wT(t)’s FT.

Units:P Watts

│WT(f)│2/T:Watts/Hz

PSD:definition

Power spectral density and autocorrelation function

26

• Definition:the PSD for a deterministic power waveform is

Pw(f)= lim[│WT(f)│2/T] Watts/Hz

Pw(f) :always a positive function of frequency. Not sensitive to the phase spectrum of w(t).

So with Pw(f),the normalized average power of w(t) can be given by:

P= -∞∫∞ Pw(f) df

Ex.

f

Pw(f)

27

Definition:the autocorrelation of a real (physical) waveform is:

Rw(τ) =<w(t)w(t+ τ)> = lim 1/T -T/2∫ T/2w(t) w(t+ τ)dt

Wiener-Khintchine theorem:

Rw(τ) Pw(f)

For P ,we have:

P =<w2(t)> =W2rms=-∞∫∞ Pw(f) df = Rw(0)

Ex. PSD of a Sinusoid

w(t)=Asinωct

the autocorrelation function:

Rw(τ) =<w(t)w(t+ τ)> =A2/2cosωc τ

so the PSD Pw(f) : Pw(f) = A2/4[δ(f+fc) +δ(f-fc)]

Autocorrelation function

28

And the average power P:

P= -∞∫∞ Pw(f) df = A2/2

f

Pw(f)

A2/4A2/4

fc-fc

29

• Complex Fourier series

w(t)=Σcnej2πf0

t a<t< T0

and

cn=1/T0 a∫a+T0 w(t) e-j2πf

0tdt

• Quadrature Fourier series (see p.70)

• Polar Fourier Series(see p.70)

Fourier series

30

• Theorem: If a waveform w(t) is periodic with period T0,the spectrum of w(t) is

W(f)= Σcn δ(f-nf0)

f0=1/ T0 and cn is the coefficients of w(t)’s complex Fourier series.

Proof. (delta function’s properties)

Line spectra for periodic waveforms

W(f)

c0

c1

c2

c3

f

c-1c-2

c-3

3f02f0f00-f0-2f0-3f0

31

• Conclusion: periodic function line spectra

no dc value c0=0

no periodic components continuous spectra

• Power spectral density for periodic waveforms

Theorem:for a periodic waveform,the PSD is given by:

Pw(f)=Σ│cn │2δ(f-nf0)

where{ cn} are the corresponding Fourier coefficients.

32

• Linear system superposition law

y(t)=L[a1x1(t)+ a2x2(t)]= a1L[x1(t)]+a2L[x2(t)]

input output

X(f) spectrum Y(f)

Rx(τ) autocorrelation Ry(τ)

Px(f) PSD Py(f)

Review of linear systems

x(t) y(t)

h(t) H(f)

33

• Impulse response

Input Output

Dirac delta function δ(t) Impulse response h(t)

x(t) y(t)=x(t)*h(t)

Transfer function

H(f)=F[h(t)]

Py(f)=│H(f)│2 Px(f)

so power transfer function of a linear network:

Gh(f) = Py(f) /Px(f) = │H(f)│2

34

• Ex. RC low-pass filter

• Sol. Using Kirchhoff’s voltage law,we have:

• Where

• FT of this differential equation:

x(t) y(t)i(t)

)()()( tytRitx

dt

tdyCti

)()(

)()()()2( fXfYfYfjRC

35

• The transfer function:

• And the impulse response of RC LPF:

• The power transfer function:

• Where

fRCjfX

fYfH

)2(1

1

)(

)()(

)(th 0,1

0/

0

te t

20

2

)/(1

1)()(

fffHfGh

00

1)2/(1

RCf

36

• Distortionless communication systems(channels): the output of the channel is just proportional to a delayed version of the input

• so we have: y(t)=Ax(t-Td)

A:gain;Td:time delay

we can represent the transfer function of such system by:

Y(f)=AX(f)e-j2πf Td

and

H(f)=Y(f)/X(f)=A e-j2πf Td

Implication:(a linear time-invariant and distortionless system)

1.│H(f)│=constant =A :the magnitude spectrum is constant.

2. Phase response is a linear function of frequency

Distortionless transmission

37

• When │H(f)│=constant =A no amplitude distortion

• When θ(f)= -2πf Td no phase distortion

• Time delay of the system is defined by:

Td(f)= -(1/2πf) θ(f)

so no phase distortion is equivalent to:

Td(f)=constant

38

Ex.Distortion caused by a RC LP filter

H(f)=Y(f)/X(f)=1/[1+j(2πRC)f]

the amplitude response is │H(f)│=1/[1+(f/f0)]1/2

Where f0= 1/(2πRC)

and the phase response is θ(f)= -arctg(f/f0)

the corresponding time delay function is

Td(f)=1/(2πf) arctg(f/f0)

Effect?

x(t) y(t)i(t)

39

• In communication systems,often we deal with signals and noise called bandlimited.

• Definition:a waveform w(t) is said to be (absolutely) bandlimited to B Hz if

W(f)=F[w(t)]=0 for │f│≥B

• Definition: a waveform w(t) is (absolutely) timelimited if

w(t)=0 for │t│≥T

• Theorem:an absolutely bandlimited waveform cannot be absolutely timelimited and vice versa.

• Dilemma :physically realizable waveform timelimited

so not bandlimited.

How to do?

Bandlimited signals and noise

40

• Hilbert Transform

• Definition:The Hilbert transform of f(t) is defined as:

• Where f(t) is real.

• The inverse Hilbert transform is:

• We have:

d

t

ftfHtf

)(1)]([)(

d

t

ftfH

)(1

)]([1

ttftf

1

)()(

41

• Spectra:

• So H(f) can be calculated by:

f(t) tth

1

)(

ttftf

1

)()(

)sgn()]([F)( 0,0, fjthfH fj

fj

42

• Properties of Hilbert transform:

If f(t) is even ， then is odd.

If f(t) is odd ， then is even .

)(])([1 tftfH

)()]([ tftfH

dttfdttf )()( 22

)(tf

)(tf

0)()(

dttftf

43

• Analytic signals

• Definition: 实信号 f(t) ，则定义复信号Z(t):

为 f(t) 的解析信号。性质：

)()()( tjftftZ

0),(20,0

*

)()()(),()(

)]()([2

1)(

)](Re[)(

ffFffZfZtzfFtf

tztztf

tztf

44

• 性质（续）：

fz

fffF

ftjftj

EE

tztztztz

tztz

tzF

dfefFdfefFtz

2

0)()(,0)()(

)((),(

)]([

)(2)(2)(

2*

1*

21

21

0,00),(2

*

0

2

0

2

解析信号的能量：

解析信号）

45

• Ex. 解析信号的确定• 方法：时域：虚部是实部的 Hilbert 变换 频域： z(t) 的 Fourier 变换是其实

部 的 Fourier 变换的两倍 若复信号的 Fourier 在 f<0 时恒为

0

故可验证 是解析信号

tfje 02

tfje 02

46

• 已知实函数 f(t) ，求其解析信号1. Hilbert 变换2. 求 F(f)

• Ex. 求 的解析信号。• 解：• 因此

0

2)(2)( dfefFtz ftj

ttf 0cos)(

)()]()([2

12cos 000 fFfffftf

tfjftj edfefffftz 02

0

200 ])([

2

12)(

47

• 频带信号与带通系统

48

• Bandpass signaling:

• 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).

49

• 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

50

• 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[.].

See pages231-232 for g[.]

51

• Bandpass signal’s spectrum complex envelope’s spectrum

• 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

52

• 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

53

• 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

54

• 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)

55

• 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

56

• 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

57

• 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

58

• 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.

• 1. Absolute bandwidth

• 2. 3-dB bandwidth (half-power)

• 3. Equivalent noise bandwidth

• 4. Null-to-null bandwidth (or zero-crossing bandwidth)

• 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