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Lecture 4
Goals:
� Be able to determine bandwidth of digital signals
� Be able to convert a signal from baseband to passband and back
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Bandwidth of Digital Data Signals
A digital data signal is modeled as a random process Y � t � which is a stationary(wide sense) version of a process X � t � ;
Y � t � � X � t� U �
where U is a random variable needed inorder to make Y � t � wide sensestationary. In many digital communication systems X � t � is an infinitesequence j pulses or waveforms i.e.
X � t � �∞
∑�� � ∞
b � x � t� � T ��
In this case if U is uniformly distributed between 0 and T then Y � t � is a widesnese stationary random process. We desire then to compute the autocorrelation of Y � t � and also the spectrum of Y � t � . Assume that b � � ∞
�� � ∞ is asequences of i.i.d. random variables with zero mean and variance σ2 (e.g.
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P b � � � 1 � � 1 � 2 P b � � � 1 � � 1 � 2). Also assume U and b � areindependent.
Claim:
RY � τ � � σ2
T
∞
� ∞x � t � x � t � τ � dt
SY � ω � � 1T � F � ω � �
2 where F � ω � � F x � t � �
Derivation: For any t and τ
E � Y � t � Y � t � τ � � � E∞
∑
�� � ∞b � x � t� � T� U �
∞
∑m� � ∞
bmx � t � τ� mT� U �
�
∞
∑
�� � ∞
∞
∑m� � ∞
E b � bm �� �� �
δ m�
σ2 � m0 �� m
E � x � t� � T� U � x � t � τ� mT� U � �
�
∞
∑
�� � ∞σ2E � x � t� � T� U � x � t � τ� � T� U � �
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�
∞
∑
�� � ∞σ2 1
T
T
u� 0x � t� � T� u � x � t � τ� � T� u � du
� 1T
∞
∑
�� � ∞σ2 � � � 1 � T
� Tx � t� v � x � t � τ� v � dv � v � � T � u dv � du �
� σ2
T
∞
� ∞x � t� v � x � t � τ� v � dv � w � t� v �
� σ2
T
∞� ∞
x � w � x � w � τ � dw � σ2
T
∞
� ∞x � t � x � t � τ � dt
Thus Y � t � is wide sense stationary with
RY � τ � � σ2
T
∞� ∞
x � t � x � t � τ � dt
Now let f1 � t � � x � t � f2 � t � � x �� t � then
f1 � f2 � τ � �
∞
� ∞f1 � t � f2 � τ� t � dt �
∞� ∞
x � t � x � t� τ � dt
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�
∞
� ∞x � t � τ � x � t � dt
So
RY � τ � � σ2
T � f1 � f2 � � τ �
SY � ω � � Fσ2
T � f1 � f2 � � τ � � σ2
TF1 � ω � F2 � ω �
F1 � ω � � F x � t � � � F � ω �
F2 � ω � � F x �� t � � � F �
� ω �SY � ω � � σ2
T � F � ω � �
2
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Example
x � t � � pT � t � ��
��
1 0 � t � T
0 elsewhere
b ��� �� 1 �� σ2 � 1.
∞
� ∞x � t � x � t � τ � dt �
T
0pT � t � τ � dt �
��
�
T� τ 0 � τ � T
T � τ � T � τ � 0
0 elsewhere
pT � t � τ � �
��
�
1 0 � t � τ � T
0 elsewhere�
��
�1 � τ � t � T� τ
0 elsewhere
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RY � τ � �
��
�
1T � T� � τ � � � τ � � T
0 elsewhere
SY � ω � � 1T � F pT � t � � �
2
F � ω � � F pT � t � � � T
�
sinωT � 2ωT � 2
�
e
� jωT
�
2
� F � ω � �
2 � T 2 sin2 ωT � 2� ωT � 2 � 2
SY � ω � � 1T � F � ω � �
2 � Tsin2 ωT � 2
� ωT � 2 � 2
� T sinc2
SY � f � � Tsin2 π f T
� π f T � 2
� T sinc2 � f T �
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Spectrum for Rectangular Pulses
−4 −3 −2 −1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
frequency
Spe
ctru
m
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Spectrum for Rectangular Pulses
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−30
−25
−20
−15
−10
−5
0
frequency
SX(f
) (d
B)
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Spectrum for Rectangular Pulses
−8 −6 −4 −2 0 2 4 6 8−40
−35
−30
−25
−20
−15
−10
−5
0
frequency
SX(f
) (d
B)
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Definition of Bandwidth for Digital Signals
1. Null-to-Null bandwidth∆� bandwidth of main lobe of power spectral
density
2. 99% power bandwidth containtment∆� bandwidth such that 1/2% of
power lies above upper band limit and 1/2% lies below lower band limit
3. x dB bandwidth∆� bandwidth such that spectrum is xdB below spectrum
at center of band (e.g. 3dB bandwidth)
4. Noise bandwidth∆� WN � P � S � fc � where P is total power and S � fc � is
value of spectrum at f � fc.
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�
�
S � f �
fc f
(Area under curve = P)
�� WN
5. Gabor bandwidth∆� σ where σ2 � �
∞� ∞ � f � fc �
2S � f � d f
�
∞� ∞ S � f � d f
6. Absolute bandwidth∆� WA � min W : S � f � � 0� � f ��� U �
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Rectangular Pulse Example
SY � f � � Tsin2 π f T
� π f T � 2
� T sinc2 � f T �
sinc � π � f� fc � T � 0 at π � f� fc � T � hπ h � � 1
�� 2
�� 3
�
� � �
f � fc � nT �
n � � 1�� 2
�
� � �Null to null bandwidth
∆� width of main lobe of spectral density.
For PSK null to null bandwidth = 2T
Fractional Power containment Bandwidth∆� width of frequency band which
leaves 1/2% of singal power above upper band limit and 1/2% of signal powerbelow band limit
For PSK 99% energy bandwidth = 20 � 56T
We would like to find modulation schemes which decrease the bandwidthwhile retaining acceptable performance
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Modulation 1 2 3 35dB 4 5 3 3dB
BPSK 2.0 20.56 35.12 1.00 ∞ 0.88
QPSK 1.0 10.28 17.56 0.50 ∞ 0.44
MSK 1.5 1.18 3.24 0.62 (0.5) 12 0.59
These three modulation schemes all have same error probability.
MSK has minimum possible Gabor bandwidth over all modulation schemes
whose basic pulse is limited to 2T seconds.
All of these have infinite absolute bandwidth.
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Shannon’s theorem revisited
Theorem: (Shannon) For a white Gaussian noise channel there exist signals
with absolute bandwidth W conveying R � 1T bits of information per second
with arbitrarily small error probability provided
R � 1T
� W log2 � 1 �
Eb
WT N0
�
where Eb � PT is the energy per data bit and P is the power of the signal
P � limT � ∞
12T
T
� Ts2 � t � dt
R � W log 1 �
RW
Eb
N0
�P
N0 ln2as W � ∞
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R � W � log 1 �
RW
Eb
N0
� Eb � N0�2R
�
W� 1R � W
� ln2 asRW
� 0
Eb � N0 � ln � 2 �� � Eb � N0 � dB � 10log10 � ln � 2 � � � � 1� 6dB
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Up and Down Conversion
In communication systems typically the signal are generated at baseband andthen up converted to the desired carrier frequency. At the receiver this processis reversed.
�
xc � t � yc � t �
cos � 2π fct �
yc � t � � xc � t � cos � 2π fct �
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Yc � f � � 12 � Xc � f� fc � � Xc � f � fc � �
Thus multiplication in the time domain by cos � 2π fct � shifts the spectrum up
and down by fc and reduces each part by 1/2.
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Re � Xc � f � � , ImXc � f �
fW� W
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Im � Xc � f � �
f
W
� W
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Re � Yc � f � � , Im � Yc � f � �
ffc� fc
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Im � Yc � f � �
ffc
fc� W
fc� W
� fc
� fc� W
� fc� W
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Now consider multiplication by� sin � 2π fct � .
�
xs � t � ys � t �
� sin � 2π fct �
ys � t � � � xs � t � sin � 2π fct �
Yc � f � � j2 � Xc � f� fc �� Xc � f � fc � �
Thus multiplication by� sin � 2π fct � shifts the spectrum up and down also
except that the real part becomes the imaginary part and the imaginary part is
inverted and becomes the real part in addition to a reduction by 1/2.
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Re � Xs � f � �
fW� W
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Im � Xs � f � �
fW
� W
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Re � Ys � f � �
ffc
fc � W
fc � W� fc� fc � W
� fc � W
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Im � Ys � f � �
ffcfc � Wfc � W� fc
� fc � W� fc � W
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Now consider adding these two functions together.
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�
xc � t � yc � t �
cos � 2π fct �
�
xs � t � ys � t �
� sin � 2π fct �
�
y � t �
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y � t � � yc � t � � ys � t �
� xc � t � cos � 2π fct �� xs � t � sin � 2π fct �
� xe � t � cos � 2π fct � θ � t � �
The signal xe � t � is called the envelope and θ � t � is called the phase.
Y � f � � Yc � f � � Ys � f �
θ � t � � tan� 1
�
xs � t �
xc � t ��
xe � t � � � x2c � t � � x2
s � t � � 1
�
2
y � t � � Re � � xc � t � � jxs � t � � e j2π fct�
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Re � Y � f � �
ffcfc � Wfc � W� fc� fc � W� fc � W
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Im � Y � f � �
ffcfc � W
fc � W
� fc� fc � W
� fc � W
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Signal Decomposition
The signals xc � t � and xs � t � can be recovered from y � t � by mixing down to
baseband and filtering out the double frequency terms. Note that we need the
exact phase of the local oscillators to do this perfectly.
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y � t �
��
2cos � 2π fct �
� 2sin � 2π fct �
GLP � f �
GLP � f �
zs � t �
zc � t �xs � t �
xc � t �
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GLP � f � is an ideal low pass filter with transfer function GLP � f � � 1 � f � � W
and GLP � f � � 0 otherwise.
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GLP � f �
f
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Consider the spectrum of zc � t � . This is given by
Zc � f � � Y � f� fc � � Y � f � fc ��
Similarly the spectrum of zs � t � is
Zs � f � � j � Y � f� fc �� Y � f � fc � ��
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Re � Zc � f � �
f2 fcW� W� 2 fc
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Im � Zc � f � �
fW
� W2 fc� 2 fc
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Re � Zs � f � �
f2 fc
W� W
� 2 fc
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Im � Zs � f � �
fW
� W2 fc
� 2 fc
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Example
xc � t � � ac
�
1 cos � 2π f1t � � ac
�
2 sin � 2π f1t � � ac
�
3 cos � 2π f2t �
xs � t � � as
�
1 cos � 2π f1t � � as
�
2 sin � 2π f1t � � as
�
3 sin � 2π f2t �
where
ac
�
1 � 0� 25
�
ac
�
2 � 0� 5
�
ac
�
3 � 1
�
as
�
1 � � 1� 0
�
ac
�
2 � 0� 25
�
ac
�
3 � 1
�
f1 � 1�
f2 � 2�The signals are upconverted with a quadrature modulator to produce
� xc � t � cos � 2π fct �� xs � t � sin � 2π fct �
� xe � t � cos � 2π fct � θ � t � �
where fc � 16.
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
time
xc(t
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
xs(t
)
time
Figure 28: Example of Up and Down Conversion of Sig-
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
time
xe(t
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-2
0
2
4
phi.(
t)
Figure 29: Example of Up and Down Conversion of Sig-
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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
xc(t)
xs(t
)
Figure 30: Real and Imaginary Part of Complex Signal
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
time
y(t)
Figure 31: Example of Up and Down Conversion of Sig-
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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
zc(t
)
time
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
zs(t
)
time
Figure 32: Example of Up and Down Conversion of Sig-
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Now consider what happens if there is an imperfect local oscillator
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y � t �
��
2cos � 2π fct � φ �
� 2sin � 2π fct � φ �
GLP � f �
GLP � f �
zs � t �
zc � t �ws � t �
wc � t �
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wc � t � � xc � t � cos � φ � � xs � t � sin � φ �
ws � t � � � xc � t � sin � φ � � xs � t � cos � φ �
wc � t � � jws � t � � � xc � t � � jxs � t � � e
� jφ
Note that this is equivalent to a phase rotation by angle φ.
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We can recover the orignial signal by rotating the signal.
xc � t � � jxs � t � � � wc � t � � jws � t � � e � jφ
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wc � t �
ws � t �
��
��
��
cos � φ �
sin � φ �
cos � φ �
�
xc � t �
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Binary Phase Shift Keying (BPSK)
�
b � t �� 2Pcos � 2π fct �
Modulator�
�
s � t �
n � t ��
�
r � t �
b � t � �
∞
∑l� � ∞
bl pT � t� lT � �
bl� � 1
�
� 1 ��
pT � t � ��
��
1
�
0 � t � T
0
�
otherwise.
�
�t
1
pT � t �T
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s � t � � � 2P∞
∑l� � ∞
bl cos � 2π fct � pT � t� lT �
� � 2P b � t � cos � 2π fct � � � 2Pcos � 2π fct � φ � t � �
where φ � t � is the phase waveform. The signal has power P. The energy of the
transmitted bit is E � PT .
The phase of a BPSK signal can take on one of two values as shown below.
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0 1 2 3 4 5 6
-1
0
1
b(t)
time
0 1 2 3 4 5 6
0
2
4ph
i(t)
time
0 1 2 3 4 5 6
-1
0
1
s(t)
time
Figure 33: Signals for BPSK Modulation
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Demodulator
�
r � t �
2 � T cos � 2π fct �
�
� LPF
�� �
X � iT �
t � iT
� 0 dec bi � 1 � � 1
� 0 dec bi � 1 � � 1
The low pass filter (LPF) is a filter “matched” to the baseband signal being
transmitted. For BPSK this is just a rectangular pulse of duration T . The
impulse response is h � t � � pT � t ��
X � t � �
∞
� ∞2 � T cos � 2π fcτ � h � t� τ � r � τ � dτ
X � iT � �
∞
� ∞2 � T cos � 2π fcτ � pT � iT� τ � r � τ � dτ
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�iT
� i � 1 � T2 � T cos � 2π fcτ �
�
� 2P b � τ � cos � 2π fcτ �
� n � τ � � dτ
�
iT� i � 1 � T2 P � T bi � 1 cos � 2π fcτ � cos � 2π fcτ � dτ � ηi
ηi is Gaussian random variable, mean 0 variance N0 � 2. Assuming
2π fcT � 2πn
X � iT � � � PT bi � 1 � ηi � � E bi � 1 � ηi�
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X(iT)
P
Pe,+1
e,-1
E E0
Figure 34: Probability Density of Decision Statistic for BinaryPhase Shift Keying
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Bit Error Probability of BPSK
Pe
�
b � Q2EN0
� Q2Eb
N0
where
Q � x � �
∞
x
12π
e
� u2
�
2du
For binary signals this is the smallest bit error probability, i.e. BPSK are
optimal signals and the receiver shown above is optimum (in additive white
Gaussian noise). For binary signals the energy transmitted per information bit
Eb is equal to the energy per signal E. For Pe�
b � 10 � 5 we need a bit-energy,
Eb to noise density N0 ratio of Eb � N0 � 9� 6dB. Note: Q � x � is a decreasing
function which is 1/2 at x � 0. There are efficient algorithms (based on Taylor
series expansions) to calculate Q � x � . Since Q � x � � e �
� x2
�
2
� � 2 the error
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probability can be upper bounded by
Pe
�
b �
12
e �
� Eb �
N0 �
which decreases exponentially with signal-to-noise ratio.
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161412108642010 -10
Error Probability of BPSK
Eb/N 0 (dB)
Pe,b
10 -9
10 -8
10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
1
Figure 35: Error Probability of BPSK.
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Bandwidth of BPSK
The power spectral density is a measure of the distribution of power withrespect to frequency. The power spectral density for BPSK has the form
S � f � � PT2 �
sinc2
� � f� fc � T � � sinc2
� � f � fc � T � �
where
sinc � x � � sin � πx �
πx
�
Notice that ∞� ∞
S � f � d f � P�The power spectrum has zeros or nulls at f� fc � i � T except for i � 0; that isthere is a null at f� fc � � 1 � T called the first null; a null at f� fc � � 2 � T
called the second null; etc. The bandwidth between the first nulls is called thenull-to-null bandwidth. For BPSK the null-to-null bandwidth is 2 � T . Noticethat the spectrum falls off as � f� fc � 2 as f moves away from fc. (The
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spectrum of MSK falls off as the fourth power, versus the second power for
BPSK).
It is possible to reduce the bandwidth of a BPSK signal by filtering. If the
filtering is done properly the (absolute) bandwidth of the signal can be
reduced to 1 � T without causing any intersymbol interference; that is all the
power is concentrated in the frequency range� 1 � � 2T � � � f� fc � � 1 � � 2T � .The drawbacks are that the signal loses its constant envelope property (useful
for nonlinear amplifiers) and the sensitivity to timing errors is greatly
increased. The timing sensitivity problem can be greatly alleviated by filtering
to a slightly larger bandwidth� � 1 � α � � � 2T � � � f� fc � � � 1 � α � � � 2T � .
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0.00
0.10
0.20
0.30
0.40
0.50S(
f)
-4 -3 -2 -1 0 1 2 3 4
(f-fc)T
Figure 36: Spectrum of BPSK
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-100
-80
-60
-40
-20
0
S(f)
(dB
)
-5 -4 -3 -2 -1 0 1 2 3 4 5
(f-f c)T
Figure 37: Spectrum of BPSK
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-100
-80
-60
-40
-20
0
S(f)
(dB
)
-10 -8 -6 -4 -2 0 2 4 6 8 10
(f-fc)T
Figure 38: Spectrum of BPSK
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Example
Given:
� Noise power spectral density of N0 � 2 � � 150 dBm/Hz =10 � 18 Watts/Hz.
� Pr � 3 � 10 � 10 Watts
� Desired Pe � 10 � 7.
Find: The data rate that can be used and the bandwidth that is needed.
Solution: Need Q � 2Eb � N0 � � 10 � 7 or Eb � N0 � 11� 3dB or Eb � N0 � 13� 52.
But Eb � N0 � PrT � N0 � 13� 52. Thus the data bit must be at least
T � 9� 0 � 10 � 8 seconds long, i.e. the data rate 1 � T must be less than 11
Mbits/second. Clearly we also need a (null-to-null) bandwidth of 22 MHz.
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An alternative view of BPSK is that of two antipodal signals; that is
s0 � t � � � Eψ � t � �
0 � t � T
and
s1 � t � � � � Eψ � t � �
0 � t � T
where ψ � t � � 2 � T cos � 2π fct � �
0 � t � T is a unit energy waveform. The
above describes the signals transmitted only during the interval � 0 �
T � .
Obviously this is repeated for other intervals. The receiver correlates with
ψ � t � over the interval � 0 �
T � and compares with a threshold (usually 0) to make
a decision. The correlation receiver is shown below.
�
r � t �
�
�
ψ � t �
�
T0 �
� γ dec s0
� γ dec s1
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This is called the “Correlation Receiver.” Note that synchronization to the
symbol timing and oscillator phase are required.
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Spectrum for Passband Pulses
Now Let Z � t � � Y � t � cos � ωct � θ � with θ uniform on [0,2π] and independentof Y � t � . Then
RZ � τ � � 12
RY � τ � cosωcτ
SZ � ω � � 14 � SY � ω� ωc � � SY � ω � ωc � �
If Z � t � � Y1 � t � cos � ωct � θ1 � � Y2 � t � cos � ωct � θ2 � with Y1 � t � and Y2 � t �
independent then
RZ � τ � � 12 � RY1 � τ � � RY2 � τ � � cosωcτ
Application: for PSK x � t � � ApT � t � . The spectrum is
SZ � f � � A2T4 �
sinc2 � f� fc � T � sinc2 � f � fc � T ��
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