EGR 544 Communication Theory
5. Characterization of Communication
Signals and Systems
Z. Aliyazicioglu
Electrical and Computer Engineering Department
Cal Poly Pomona
Modulation Principles
• Almost all communication systems transmit digital data using s
sinusoidal carrier waveform
• The transmitted channel has limited band-width and
– is centered about the carrier for double side modulation
– is next to carrier signal for single side.
Physical modulation system implementation
– Process digital information at baseband
– Pulse shaping and filtering of digital waveform
– Mixing with carrier signal
– Filtering RF signal and amplifying to transmission
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 2
1
Modulation Signals representation
We can modify amplitude, phase , or frequency of baseband signal
Amplitude Shift Keying (ASK) or On/Off Keying (OOK)
1 ⇒ A cos(2π f ct )
0 ⇒ 0
Frequency Shift Keying (FSK)
1 ⇒ A cos(2π f1t )
0 ⇒ A cos(2π f 2t )
Phase Shift Keying (PSK)
1 ⇒ A cos(2π f ct )
0 ⇒ A cos(2π f ct + π ) = − A cos(2π f ct + π )
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 3
Representation of Band-Pass Signal
• The transmitted signal is usually a real valued band-pass signal
and let’s call s(t)
• Mathematical model of a real-valued narrowband band-pass
signal is
S ( f ) ≠ 0 for f c − f B ≤ f ≤ f c + f B and f c f B
S( f )
½ S-(f)=u(-f)S(f)
½ S+(f)=u(f)S(f)
fc
-fc
f
S(f) is Fourier transform of s(t). u(f) is the unit step function in
frequency domain
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 4
2
Representation of Band-Pass Signal
Goal is to develop a mathematical representation in time domain of
S+(f ) and S- (f )
The time domain representation of S+(f ) is s+(t), which is called
pre-envelope of s(t)
∫
s+ (t ) =
=
∞
−∞
S+ ( f )e j 2π ft df
∞
∫ [ 2u ( f ) S ( f ) ] e
j 2π ft
−∞
df
= F −1 [2u( f )] ∗ F −1 [ S ( f )]
where
j

= δ (t ) +  ∗ s(t )
πt 

j
= s (t ) + ∗ s ( t )
πt
= s(t ) + jsˆ(t )
Cal Poly Pomona
sˆ(t ) =
=
1
πt
1
∗ s (t )
π∫
∞
−∞
s(τ )
dτ
t −τ
Electrical & Computer Engineering Dept.
EGR 544-5 5
Representation of Band-Pass Signal
sˆ(t ) may be considered the output of the filter such as
s(t )
1
πt
sˆ(t )
Hilbert Transform
• The frequency response of the filter is
∞
H ( f ) = ∫ h(t )e − j 2π ft dt =
−∞
− j

H ( f ) = − j sgn( f ) =  0
 j

Cal Poly Pomona
1
1 − j 2π ft
e
dt
π ∫−∞ t
∞
f > 0 H ( f ) = 1 for f ≠ 0
f =0
 π
f <0
 − 2 for f > 0
Θ( f ) = 
 π for f < 0
 2
Electrical & Computer Engineering Dept.
Fourier Transform of
Hilbert Transform
 j
F   = sgn( f )
π t 
1
F   = − j sgn( f )
π t 
EGR 544-5 6
3
Representation of Band-Pass Signal
s(t )
sˆ(t )
1
πt
s+ (t )
+
X
j
Hilbert Transform
S( f )
S+(f)=2u(f)S(f)
f
fc
-fc
Sl ( f ) = S + ( f + f c )
f
The low-pass representation of S+(f ) is Sl ( f ) = S+ ( f + f c ) in
time domain
sl (t ) = s+ (t )e − j 2π fct = [ s(t ) + jsˆ(t )] e − j 2π fct
In complex form
s(t ) = x (t ) cos(2π f ct ) − y (t )sin(2π f ct )
sˆ(t ) = x (t )sin(2π f ct ) + y (t ) cos(2π f ct )
sl (t ) = x(t ) + jy (t )
x(t) and y(t):quadrature components of sl(t)
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 7
Representation of Band-Pass Signal
• Another representation of the signal
{
s(t ) = Re [ x (t ) + jy (t )] e j 2π fct
}
= Re  sl (t )e j 2π fct 
Or we can represent
sl (t ) = a (t )e jθ ( t )
a ( t ) = x 2 ( t ) + y 2 (t )
θ (t ) = tan −1
y (t )
x (t )
s(t ) = Re  sl (t )e j 2π fct 
= Re  a (t )e jθ ( t ) e j 2π fct 
= a (t ) cos [2π f ct + θ (t )]
a(t) is called envelope of s(t) and θ(t) is called the phase of s(t)
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 8
4
Representation of Band-Pass Signal
• The energy in the signal s(t) is defined as
∞
∞
−∞
−∞
ε = ∫ s 2 (t ) dt = ∫
{Re  s (t )e
l
j 2π f c t
}

2
dt
• Using representation of s(t) in cosine form
∞
∞
−∞
−∞
ε = ∫ s 2 (t ) dt = ∫
Then
ε=
{a(t ) cos [2π f t + θ (t )]}
2
c
dt
1 ∞ 2
1 ∞
a (t ) dt + ∫ {a 2 (t ) cos [4π f ct + 2θ (t )]} dt
∫
−∞
2
2 −∞
a(t) is the envelope and varies slowly relative to cosine function
ε=
Cal Poly Pomona
1 ∞ 2
1 ∞
a (t ) dt = ∫ sl2 (t ) dt
∫
−∞
2
2 −∞
Electrical & Computer Engineering Dept.
EGR 544-5 9
Representation of Linear Band-Pass signal
•
•
A linear filter or system can be represented either h(t) or of H(f).
Since h(t) is real
H ( f ) = H * (− f )
And Hl*( - f – fc )
Let’s define Hl( f – fc )
H ( f ) f > 0
Hl ( f − fc ) = 
f <0
0
Then, we have
0 f >0

H l* ( − f − f c ) =  *
H
(
f <0
−f)

H ( f ) = H l ( f − f c ) + H l* ( − f − f c )
The Inverse transform of H(f)
h(t ) = hl (t )e j 2π fct + hl* (t )e − j 2π fct
= 2 Re  hl (t )e j 2π fct 
hl(t), inverse transform of Hl(f), is impulse response of low-pass system
and is comlex
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 10
5
Response of Band-pass System to a Band-pass signal
s(t)
r(t)
h(t)
Let’s have
•s(t) narrowband band-pass signal and is the equivalent low-pass
signal sl (t) .
•Band-pass filter (system) the impulse response h(t) and its
equivalent low-pass impulse response hl (t)
The output of the band-pass filter is r(t) also band-pass signal
r (t ) = Re  rl (t )e j 2π fct 
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 11
Response of Band-pass System to a Band-pass signal
r(t) can be given as
∞
r (t ) = ∫ s(τ )h(t − τ )dτ
−∞
In frequency domain
R( f ) =
R( f ) = S ( f ) H ( f )
1
 Sl ( f − f c ) + Sl* ( − f − f c )  H l ( f − f c ) + H l* ( − f − f c ) 
2
For narrow band signal s(t) and narrow band system h(t)
Sl ( f − f c ) H l* ( − f − f c ) = 0
R( f ) =
R( f ) =
Sl* ( − f − f c ) H l ( f − f c ) = 0
1
 Sl ( f − f c ) H l ( f − f c ) + Sl* ( − f − f c ) H l* ( − f − f c ) 
2
1
 Rl ( f − f c ) + Rl* ( − f − f c ) 
2
Rl ( f ) = Sl ( f ) H l ( f )
∞
rl (t ) = ∫ sl (τ )hl (t − τ )dτ
−∞
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 12
6
Bandpass Stationary Stochastic Process
Let’s have n(t) as narrowband band-pass process with spectral
density is much smaller than fc , zero mean and power
spectral density Φnn(f).
And we can represent it as
n(t ) = a (t ) cos [2π f ct + θ (t )]
= x(t ) cos(2π f ct ) − y (t )sin(2π f ct )
= Re  z (t )e j 2π fct 
a(t) is the envelope and θ(t) is the phase of the real valued
signal. x(t) and y(t) are the quadrature component of n(t).
z(t) is the complex envelope of n(t)
n(t) is zero mean, therefore x(t) and y(t) will be zero mean , The
autocorrelation and cross-correlation function satisfy
φ xx (τ ) = φ yy (τ )
φ xy (τ ) = −φ yx (τ )
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 13
Bandpass Stationary Stochastic Process
• The autocorrelation function φnn(τ) of n(t)
φnn (τ ) = E[n(t )n(t + τ )] = E {[ x(t ) cos 2π f ct − y (t )sin 2π f ct ]
[ x(t + τ ) cos 2π f c (t + τ ) − y (t + τ )sin 2π f c (t + τ )] }
φnn (τ ) = φ xx (τ ) cos 2π f ct cos 2π f c (t + τ ) + φ yy (τ )sin 2π f ct sin 2π f c (t + τ )
− φ xy (τ )]sin 2π f ct cos 2π f c (t + τ ) − φ yx cos 2π f ct sin 2π f c (t + τ )
• Using trigonometric identities
1
1
2
2
1
1
− [φ yx (τ ) − φ xy (τ )]sin 2π f cτ − [φ yx (τ ) + φ xy (τ )]sin 2π f c (2t + τ )
2
2
φnn (τ ) = [φ xx (τ ) + φ yy (τ )]cos 2π f cτ + [φ xx (τ ) − φ yy (τ )]cos 2π f c (2t + τ )
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 14
7
Bandpass Stationary Stochastic Process
• Using zero mean properties
φnn (τ ) = φ xx (τ ) cos 2π f cτ + φ yx (τ )sin 2π f cτ
• z(t) can be given complex-valued as z(t)=x(t)+jy(t)
• The autocorrelation function of z(t) is given as
1
2
1
= [φ xx (τ ) + φ yy (τ ) − jφ xy (τ ) + jφ yx (τ )]
2
= φ xx (τ ) + φ yy (τ )
φ zz (τ ) = E[ z* (t ) z (t + τ )]
• The autocorrelation function φnn(τ) of n(t) can be obtained
by φzz(τ) and the carrier frequency fc
φnn (τ ) = Re φ zz (τ )e j 2π f t 
c
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 15
Bandpass Stationary Stochastic Process
• The Fourier transform of the autocorrelation function φnn(τ) gives
The power spectral density Φnn(f)
Φ nn ( f ) =
=
∫
∞
−∞
Re φ zz (τ )e j 2π fct  e − j 2π fcτ dτ
1
[Φ zz ( f − f c ) + Φ zz (− f − f c )]
2
• Where Φzz(f) is the power density spectrum of equivalent lowpass process z(t).
• Φzz(f) is real-valued function and the autocorrelation function of
z(t) satisfy that
φzz (τ ) = φ zz* (τ )
Cal Poly Pomona
Electrical & Computer Engineering Dept.
EGR 544-5 16
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