Signals and Systems - Department of Engineering and Physics

Report
ENGR 4323/5323
Digital and Analog Communication
Ch2
Signals and Signal Space
Engineering and Physics
University of Central Oklahoma
Dr. Mohamed Bingabr
Outline
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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
Signal Energy and Power
Energy Signal
Power Signal
Energy
Power
Signal Classification
1. Continuous-time and discrete-time signals
2. Analogue 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
Analog continuous
Digital continuous
Analog Discrete
Digital Discrete
Periodic
Deterministic
Aperiodic Probabilistic
Useful Signal Operation
Time Shifting
Time Scaling
Time Inversion
Useful Signals
Unit impulse Signal
Unit step function u(t)
Signals Versus Vectors
By sampling, a continuous signal g(t) can be
represented as vector g.
g = [ g(t1) g(t2) … g(tn)]
Vector Approximation
To 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
Signals Versus Vectors
Value of c that minimizes the error
Signal Approximation
g(t) = cx(t) + e(t)
x(t)
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.
Example
Which of the signals g1(t), g2(t), …, g6(t) are similar to
x(t)?
1=1
2=1
3=-1
4=0.961
5=0.628
6=0
Correlation Functions
Cross-correlation Function
Autocorrelation Function
Orthogonal Signal Sets
Orthogonal Vector Space
g = c1x1 + c2x2 + c3x3
Orthogonal Signal Space
g(t) = c1x1(t)+ c2x2(t) + … + cNxN(t)
Parseval’s Theorem
Trigonometric Fourier Series


n 1
n 1
xt   a0   an cos2nf0t    bn sin 2nf0t 
1
a0   xt dt
T0 T0
2
an   xt  cos2nf0t dt
T0 T0
2
bn   xt sin 2nf0t dt
T0 T0
Example

f t   a0   an cos2nt   bn sin 2nt 
f(t)
1
n 1
e-t/2

0

• Fundamental period
T0 = 
• Fundamental
frequency
f0 = 1/T0 = 1/ Hz
w0 = 2/T0 = 2 rad/s
a0 
an 
1

2


0


0




2
2
e dt    e  1  0.504


e

t
2

t
2
 2 
cos2nt  dt  0.504 
2 
 1  16n 
 8n 
bn   e sin 2nt  dt  0.504 
2 
0

 1  16n 
an and bn decrease in amplitude as n  .
2


t
2

2
8n


f t   0.504 1  
cos2nt  
sin 2nt 
2
2
1  16n
 n 1 1  16n

To what value does the FS converge at the point of discontinuity?
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
xt  

C0

dc component
  Cn cos2nf0t   n 



n 1
nth harmonic
C0, Cn, and θn are related to the trigonometric
coefficients an and bn as:
C0  a0
Cn  an  bn
2
2
 bn 
 n   tan  
 an 
1
Role of Amplitude in Shaping Waveform

xt   C0   Cn cos2nf0t   n 
n 1
Role of the Phase in Shaping a
Periodic Signal

xt   C0   Cn cos2nf0t   n 
n 1
Compact Trigonometric

f t   C0   Cn cos2nt   n 
f(t)
n 1
1
a0  0.504
e-t/2

0
 2 
an  0.504
2 
 1  16n 
 8n 
bn  0.504
2 
 1  16n 
C0  ao  0.504

• Fundamental period
T0 = 
• Fundamental frequency
f0 = 1/T0 = 1/ Hz
w0 = 2/T0 = 2 rad/s

2

Cn  a  b  0.504
2
 1  16n
2
n
2
n
  bn 
   tan1 4n
 an 
 n  tan1 

f t   0.504 0.504
n 1
2
1  16n
2

cos 2nt  tan1 4n





Line Spectra of x(t)
• 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 Cn  phase(Cn )
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
f(t)
e-t/2


0

2
Cn  0.504
2
 1  16n
C0  0.504
1

f t   0.504 0.504
n 1
 n   tan1 4n
2
cos 2nt  tan1 4n
1  16n 2






f(t)=0.504 + 0.244 cos(2t-75.96o) + 0.125 cos(4t-82.87o) +
o) + 0.063 cos(8t-86.24o) + …
0.084
cos(6t-85.24
C
n
n
0.504
0.244
w
0.125
0.084
0
2
4
6
0.063
8
10
w
-/2
Exponential Fourier Series
xt  

D e
n  
j 2f 0 nt
n
 j 2f 0 nt
To find Dn multiply both side by e
over a full period, m =1,2,…,n,…
and then integrate
1
Dn   xt e j 2f 0nt dt , n  0,  1,  2,....
To To
Dn is a complex quantity in general Dn=|Dn|ej
D-n = Dn*
|Dn|=|D-n|
Even
Dn = -
D-n
Odd
D0 is called the constant or dc component of x(t)
Line Spectra in the Exponential Form
• The line spectra for the exponential form has
negative frequencies because of the
mathematical nature of the complex exponent.
x(t )  ... | D 2 | e  j 2 e  j 2w0t  | D1 | e  j1 e  jw0t  D0 
| D1 | e j1 e jw0t  | D2 | e j 2 e j 2w0t  ...
x(t )  C0  C1 cos(w0t  1 )  C2 cos(2w0t   2 )  ...
|Dn| = 0.5 Cn
Dn =
Cn
Example
Find the exponential Fourier Series for the squaref(t)
pulse periodic signal.
1
Dn 
2
 jnt
e
 dt
 / 2
sin n / 2

 0.5sinc(n / 2)
n
1
D0 
2
n even
0
Dn  
 1 / n n odd
 0
n  
 
1
 /2
for all n  3,7,11,15, 
n  3,7,11,15, 
2
 /2
/2

2
• Fundamental period
T0 = 2
• Fundamental frequency
f0 = 1/T0 = 1/2 Hz
w0 = 2/T0 = 1 rad/s
Exponential Line Spectra
|Dn|
1
1
Dn
1
1
Example
The compact trigonometric Fourier Series
coefficients for the square-pulse periodic signal.
f(t)
1
C0 
2
 0 n even
Cn   2
n odd
 n
 0 for all n  3,7,11,15,
n  
n  3,7,11,15, 
 
1
2
 /2
/2

2
Relationships between the Coefficients
of the Different Forms
an  Dn  D n  2 ReDn 
2
2
bk  j Dn  D n   2 ImDn 
Cn  an  bn
an  Cn cos n 
bn  Cn sin  n 
1  bn 
 n   t an  
a0  D0  c0
 an 
Dn  0.5an  jbn 
D n  D  n  0.5an  jbn 
Dn  0.5Cn  n  0.5Cn e j n
D0  a0  C0
Cn  2 Dn
 n  Dn
C0  a0  D0
Example
Find 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
T (t )
Dn  1 / T0
1
 T0 (t ) 
T0
0

jnw0t
e

n  
Cn  2 | Dn | 2 / T0
C0 | D0 | 1 / T0
1
 T0 (t ) 
T0



1  2 cos(nw0t )
n 1


-2T0 -T0
T0
2T0
Rectangular Pulse Train Example
Clearly x(t) satisfies the Dirichlet conditions.
x(t)
1
2
 /2
/2
The compact trigonometric form is


2

1  2


( n 1) / 2
x(t )    cos nt  (1)
1 
2 n1 n
2

n odd
Parseval’s Theorem
• Let x(t) be a periodic signal with period T
• The average power P of the signal is defined as
1
P
T

T /2
T / 2
2
x(t ) dt
• Expressing the signal as

xt   C0   Cn cos(nw0t   n )
n 1
it is also

P  C0   0.5Cn
2
n 1
2

P  D  2 Dn
2
0
n 1
2

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