engineering and physics university of central oklahoma dr. mohamed bingabr
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ENGR 4323/5323 Digital and Analog Communication. Ch2 Signals and Signal Space. Engineering and Physics University of Central Oklahoma Dr. Mohamed Bingabr. Outline. Size of a Signal Classification of Signals Useful Signals and Signal Operations Signals Versus Vectors - PowerPoint PPT PresentationTRANSCRIPT
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Engineering and PhysicsUniversity of Central Oklahoma
Dr. Mohamed Bingabr
Ch2Signals and Signal Space
ENGR 4323/5323Digital and Analog Communication
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Outline
• Size of a Signal• Classification of Signals• Useful Signals and Signal Operations• Signals Versus Vectors• Correlation of Signals• Orthogonal Signal Sets• Trigonometric and Exponential Fourier Series
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Signal Energy and Power
Energy Signal
Power Signal
Energy
Power
𝐸𝑔=∫− ∞
∞
𝑔2 (𝑡 )𝑑𝑡 𝐸𝑔=∫− ∞
∞
|𝑔 (𝑡)|2𝑑𝑡
𝑃𝑔= lim𝑇 →∞
1𝑇 ∫
−𝑇 / 2
𝑇 / 2
𝑔2 (𝑡 )𝑑𝑡 𝑃𝑔= lim𝑇 →∞
1𝑇 ∫
−𝑇 / 2
𝑇 / 2
¿𝑔 (𝑡 )∨¿2𝑑𝑡 ¿
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Signal Classification
1. Continuous-time and discrete-time signals 2. Analog and digital signals 3. Periodic and aperiodic signals 4. Energy and power signals 5. Deterministic and probabilistic signals 6. Causal and non-causal 7. Even and Odd signals
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Analog continuous Digital continuous
Analog Discrete Digital Discrete
PeriodicDeterministic Aperiodic Probabilistic
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Useful Signal Operation
Time Shifting Time Scaling Time Inversion
1-1
12
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Useful Signals
Unit impulse Signal
Unit step function u(t)𝑑𝑢(𝑡)𝑑𝑡 =𝛿(𝑡 )
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Signals Versus Vectors
By sampling, a continuous signal g(t) can be represented as vector g.
g = [ g(t1) g(t2) … g(tn)]
Vector ApproximationTo approximate vector g using another vector x then we need to choose c that will minimize the error e.
g = cx + e
Dot product: <g, x> = ||g||.||x|| cos θ
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Signals Versus Vectors
Value of c that minimizes the error
Signal Approximationg(t) = cx(t) + e(t)
x(t)
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Correlation of Signals
Two vectors are similar if the angle between them is small.
Correlation coefficient
Note: Similarity between vectors or signals does not depend on the length of the vectors or the strength of the signals.
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Example
1=12=13=-1
4=0.9615=0.6286=0
Which of the signals g1(t), g2(t), …, g6(t) are similar to x(t)?
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Cross-Correlation Function
4
g(t-)
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Autocorrelation Function
g(t)
g(t-)
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Orthogonal Signal Sets
g = c1x1 + c2x2 + c3x3
Orthogonal Vector Space
g(t) = c1x1(t)+ c2x2(t) + … + cNxN(t)
Orthogonal Signal Space
Parseval’s Theorem
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Trigonometric Fourier Series
1 1
000 2sin2cosn n
nn tnfbtnfaatx
∫0
00
2cos2
Tn dttnftx
Ta
∫0
00
2sin2
Tn dttnftx
Tb
∫00
01
T
dttxT
a
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Example
• Fundamental periodT0 =
• Fundamental frequencyf0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s
. as amplitudein decrease and 1618 504.0 2sin2
1612 504.0 2cos2
504.0121
2sin2cos
202
202
20
20
10
∫
∫
∫
nban
ndtnteb
ndtntea
edtea
ntbntaatf
nn
t
n
t
n
t
nnn
0
1e-t/2
f(t)
122 2sin
16182cos
16121504.0
n
ntn
nntn
tf
To what value does the FS converge at the point of discontinuity?
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1 harmonicnth
0
component dc
0 2cosn
nn tnfCCtx
00 aC 22
nnn baC
n
nn a
b1tan
Compact Trigonometric Fourier Series
We can use the trigonometric identity a cos(x) + b sin(x) = c cos(x + )to find the compact trigonometric Fourier series
C0, Cn, and θn are related to the trigonometric coefficients an and bn as:
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Role of Amplitude in Shaping Waveform
1
00 2cosn
nn tnfCCtx
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Role of the Phase in Shaping a Periodic Signal
1
00 2cosn
nn tnfCCtx
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Compact Trigonometric
• Fundamental periodT0 =
• Fundamental frequencyf0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s
nab
nbaC
aCn
nb
na
a
ntCCtf
n
nn
nnn
o
n
n
nnn
4tantan
161
2504.0
504.01618 504.0
1612 504.0
504.0
2cos
11
2
22
0
2
2
0
10
0
1e-t/2
f(t)
1
1
24tan2cos
161
2504.0504.0n
nntn
tf
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• The amplitude spectrum of x(t) is defined as the plot of the magnitudes |Cn| versus w
• The phase spectrum of x(t) is defined as the plot of the angles versus w
• This results in line spectra• Bandwidth the difference between the
highest and lowest frequencies of the spectral components of a signal.
Line Spectra of x(t)
)( nn CphaseC
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Line Spectra
nn
CC
n
n
4tan161
2504.0 504.0
1
20
0
1e-t/2
f(t)
1
1
24tan2cos
161
2504.0504.0n
nntn
tf
f(t)=0.504 + 0.244 cos(2t-75.96o) + 0.125 cos(4t-82.87o) + 0.084 cos(6t-85.24o) + 0.063 cos(8t-86.24o) + …
0.504
0.244
0.1250.084
0.063
Cn
w0 2 4 6 8 10
w
n
-/2
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n
ntfjneDtx 02
D-n = Dn*
,....2,1,0 , 102 ∫ ndtetx
TD
oT
ntfj
on
Exponential Fourier Series
To find Dn multiply both side by and then integrate over a full period, m =1,2,…,n,…
ntfje 02
Dn is a complex quantity in general Dn=|Dn|ej
Even Odd
|Dn|=|D-n| Dn = - D-n
D0 is called the constant or dc component of x(t)
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• The line spectra for the exponential form has negative frequencies because of the mathematical nature of the complex exponent.
Line Spectra in the Exponential Form
Cn = 2 |Dn| n 0
Dn = Cn
...)2cos()cos()(
...||||
||||...)(
2021010
2221
012
2
001
0102
ww
ww
ww
tCtCCtx
eeDeeD
DeeDeeDtxtjjtjj
tjjtjj
D0= C0
D-n = - Cn
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Example
• Fundamental periodT0 = 2
• Fundamental frequencyf0 = 1/T0 = 1/2 Hzw0 = 2/T0 = 1 rad/s
/2/2
1f(t)
22
Find the exponential Fourier Series for the square-pulse periodic signal.
∫
,15,11,7,3,15,11,7,3 allfor 0
odd /1even 0
21
)2/sinc(5.02/sin
21
0
2/
2/
nn
nnn
D
D
nn
n
dteD
n
n
jntn
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|Dn|
Dn
1 1
1 1
Exponential Line Spectra
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Example
,15,11,7,3,15,11,7,3 allfor 0
odd 2
even 021
0
nn
nn
nC
C
n
n
/2/2
1f(t)
22
The compact trigonometric Fourier Series coefficients for the square-pulse periodic signal.
1
2/)1(
21)1(cos2
21)(
n
nntn
tf
• Fundamental frequencyf0 = 1/T0 = 1/2 Hzw0 = 2/T0 = 1 rad/s
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000
sincos
Im2Re2
cDaCb
CaDDDjb
DDDa
nnn
nnn
nnnk
nnnn
Relationships between the Coefficients of the Different Forms
000
1
22
2
tan
DaCD
DC
ab
baC
nn
nn
n
nn
nnn
000
5.05.0
5.0
5.0
CaDeCCD
jbaDD
jbaD
njnnnn
nnnn
nnn
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ExampleFind the exponential Fourier Series and sketch the corresponding spectra for the impulse train shown below. From this result sketch the trigonometric spectrum and write the trigonometric Fourier Series.Solution
10
0
000
0
0
0
)cos(211)(
/1||/2||2
1)(
/1
0
0
0
nT
nn
n
tjnT
n
tnT
t
TDCTDC
eT
t
TD
w
w
2T0T0-T0-2T0
)(0
tT
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1- The function g(t) is absolutely integrable over one period.
2- The function g(t) can have only a finite number of maxima, minima, and discontinuities in one period.
Dirichlet Conditions for FS Convergence
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• Let x(t) be a periodic signal with period T• The average power P of the signal is defined as
• Expressing the signal as
it is also
1
00 )cos(n
nn tnCCtx w
1
220 2
nnDDP
1
220 5.0
nnCCP
Parseval’s Theorem
∫2/
2/
2)(1 T
Tdttx
TP
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Numerical Calculation of Fourier Series
,....2,1,0 , 102 ∫ ndtetg
TD
oT
ntfj
on
𝐷𝑛= lim𝑇 𝑠→ 0
1𝑇 0
∑𝑘=0
𝑁 0 −1
𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛𝜔 0𝑘𝑇𝑠𝑇 𝑠
𝐷𝑛= lim𝑇 𝑠→ 0
1𝑁0
∑𝑘=0
𝑁 0 −1
𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛Ω 0𝑘
𝐷𝑛=1𝑁0
∑𝑘=0
𝑁 0 −1
𝑔 (𝑘𝑇 𝑠 )𝑒− 𝑗𝑛Ω 0𝑘
Ω 0=𝜔0𝑇 𝑠
𝑁 0=𝑇 0
𝑇 𝑠
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Matlab Exercise Page 80
T0 = pi; N0 = 256; Ts = T0/N0;t = 0: Ts : Ts*(N0-1); g = exp(-t/2); g(1) = 0.604;Dn = fft(g)/N0;[Dnangle, Dnmag] = cart2pol( real(Dn), imag(Dn));k =0:length(Dn)-1; k = 2*k;subplot(211), stem(k, Dnmag)subplot(212), stem(k,Dnangle)
0
1e-t/2
g(t)
T0 = f0 = 1/T0 = 1/ Hzw0 = 2/T0 = 2 rad/s