1
2W
fs
fs
x(t)
x(t)
x(t) x( nTs )
x (t)
x (t) X (f)
X( f ) X (f)
x(t) x( nTs )
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x (t )
x (t)
x (t) x(t) (t nTs ) x (nTs ) (t nTs )
n n
x( nTs ) x(t) t nTs n 0, 1, 2, 3
x (t) x(t) t)
(t nTs ) Ts 2Ts
3Ts
x (t ) X (f)
X (f) x(t) (t nTs )
n
x(t)
X (f) {x(t)} FT { (t nTs )}
FT
x(t)
X( f )
X(f)
FT
(t nTs )
fs ( f nfs )
n Spectrum of
original signal
f
X (f) X( f ) fs ( f nfs ) –W W
n X(f)
X (f) fs X ( f ) ( f nfs )
n
fs X ( f nfs )
n
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fs X( f 2 fs ) fs X( f fs ) fs X( f ) fs X( f fs ) fs X( f 2 fs )
X(f) is placed at Spectrum of
fs, 2fs,... X(f) sampled signal
W W
f = nfs
–2fs –fs–W –fs –W W fs fs+W 2fs
–3 W 3W
fs , 2 fs , 3 fs ,
X( f )
X(f) X (f)
fs
X (f) fs X( f ) W
1
X( f ) X (f)
fs
x( t ) x( nT s )
X( ) x(n) e jn
n
X( f ) x(n) e j 2fn
n
X( f ) X (f)
f
j 2 n
X (f) x(n) e fs
n
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f
fs
x( n) x( nTs ) x(t)
1
X (f) x(nTs ) e j 2fnTs fs
Ts
n
1
X( f )
fs x(n Ts ) e j 2 fnTs
n
x(t)
1
x(t) IFT
f x(n Ts ) e j 2 f nTs
s n
x(t) x( n Ts )
fs
x(t)
x( t )
X( f ) X (f)
x(t)
1
x(t) f x(n Ts ) e j 2 f nTs e j 2 f t df
s n
1
W f W X( f ) X (f)
fs
W f W
W
1
x(t) fs x(n Ts ) e j 2 f nTs e j 2 f t df
W n
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W
1
x(t) x(n Ts ) fs e j 2 f ( t nTs ) df
n W
W
1 e j 2f ( t nTs )
x(n Ts ) fs j 2 (t n Ts )
n W
1 e j 2 W ( t nTs ) e j 2 W ( t nTs )
x(n Ts ) fs j 2 (t n Ts )
n
1 sin 2 W (t n Ts )
x(n Ts ) fs (t n Ts )
n
sin ( 2 W t 2 W n Ts )
x(n Ts ) ( fs t fs n Ts )
n
1 1
fs 2W
sin ( 2 W t n)
x(t) x( n Ts )
( 2 W t n)
n
sin
x(n Ts ) sinc (2 W t n)
n
x(t) x( 2Ts) sinc( 2 W t + 2)+ x( Ts) sinc( 2 W t 1) x( 0) sinc( 2 W t) x(Ts) sinc( 2 W t 1)
x( n Ts )
x(t)
x(t)
W f W
x(t)
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Reconstructed
signal x(t)
Sinc function at Sample of x(t) at
t = 2Ts t = 2Ts
– 4Ts – 3Ts – 2Ts – Ts 0 Ts 2Ts 3Ts 4Ts
x (t )
fs
fs 2 W X (f)
X(f) Spectrum of
original signal
This is high frequency
'W' of spectrum of
X(f–fs). But it overlaps
on X(f) and appears as
low frequency of
–W 0 W f fs–W
Overlapping X(f) These frequencies
spectrums are aliased
Spectrum of
X(f–fs)
–2fs –fs 0 W fs 2fs 0 W fs
f fs–W
–W fs–W High frequency appearing as
low frequency is called aliasing
X( f ), X( f fs ), X( f 2 fs ),
X( f ) X( f fs )
( fs W )
X( f fs ) ( fs W )
X( f )
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fs
X(f) This gap avoids
aliasing
–fs –W 0 W fs f
fs–W
fs
fs
Band limiting x(t)
x'(t) Sampler X(t)
low pass filter
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1
2W
x( n Ts )
X(f)
W f W
X(f)
This is X(f)
–fs –W 0 W fs f
H(f)
1 This response will
filter X(f) from X(f)
–W 0 W f
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x(t)
fm x(t)
x(t)
c(t) x(t) c(t)
y(t)
x(t)
Modulating y(t) Pulse PAM
Lowpass Multiplier signal
signal x(t) shaping
filter s(t)
network
c(t)
Pulse-train
generator
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x(t) (a)
c(t) (b)
y(t) t
(c)
s(t) t
(d)
^
x(t)
PAM Reconstruction Reconstructed
signal lowpass PAM signal t
filter
(a) Detector (b) Waveforms
' ' Ts
x (t)
fs
fs
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1 1
Ts fs
2W Ts
1
t Ts
2W
1 1
f
2
1
f max
2
f max
B T f max
1
BT
2
1 1
B W
2W T 2
BT W
BT
x (t)
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x(t)
(a) Modulating
signal
ts t
s(t) (b) Trailing
edge
ts tt t
s(t)
(c) Leading
edge
' ts ' tl ts t
s(t)
ts tt (d) Both(d)
edge
tl ts tt t
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ts
ts
t l and tt
+VCC
8 4
R
2 Trigger
7 IC
555 5 Modulating signal
6
C
3 Output
1
Trigger input
t
Modulating signal
Pulse width modulated output
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+VCC
RL R RL PWM
C output
Trigger in
RD C1 C2
B1
T1 T2
B2
RD
RE
RD CB
Input
signal
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PWM
signal
1
Schmitt
trigger
2 Ramp 3 5 Level 6
Adder Rectifier
generator shifter
Low 8 Demodulated
Synchronization 4 pass output
pulse generator filter
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Input
1 PAM
signal
0 t
2 Output of
schmitt trigger
t
0
Output of
3 Ramp generator
t
0
Synchronization
4
pulses
t
0
Output of
5 adder
t
0
Output of
6 level shifter
t
0
7 Output of
rectifier
t
0
Recovery
8 modulating
signal
t
0
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S()
fm
fm
Frequency
fm fs 2fs
fm fs
fs
Ts
x (t)
x (t)
kTs
x (t)
x (t)
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Comparator
X(t)
PDM/PWM
(a)
Sawtooth PPM
Monostable
generator
x(t)
Time
A
PDM (b)
Time
kTs
A
PPM
Time
kTs tk
x (t)
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kTs x (t) kTs
+VCC
8 4
R2
D1 R1
7 IC
555 3 PPM
6 output
C1
PWM 2 1 5
pulses C2
C3
PWM
pulses
t
Trigger
pulses
t
PPM
pulses
t
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PPM
pulses R Flip PWM PWM Recovered
flop Q pulses demodulator modulating
S
signal
Synchronization Reference
signal from pulse
transmitter generator
Position of unmodulated pulses
Triggering
pulses t
Output of flip-flop
(PWM pulses) t
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Ts
tr Ts
1
BT
2tr
1
BT
2tr
Waveform
Waveform Waveform
A0
0
Time Time Time
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BT
fs 8 kHz 01
. Ts
1 1
Ts Ts sec
fs 8 10 3
01
.
. Ts
01 sec
8 10 3
BT
1 1
BT
2 01
.
2
8 10 3
01
.
sec
8 10 3
01
.
tr 0.01 0.01 1.25 10 7 sec
8 10 3
1 1
BT
2tr 2 1 . 25 10 7
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x(t) sinc ( 200 t) x(t) sinc2 ( 200 t) x(t) sinc ( 200 t) sinc2 ( 200 t)
1 1 1
200 400 400
{ fm f fm}
|f| W x (nTs) xq(nTs)
Pre alias Sampling Quantization Encoding PCM
x(t)
LPF circuit circuit circuit signal
1
fs 2W Ts
fs
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PCM Regenerator Digital to Low pass
signal Analog converter x (t)
circuit filter
q 2v
fs
fs
r v fs
fs 2W
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B 1 r (5.2.4)
T 2
1
B T 2 v fs Since fs 2 W (5.2.5)
B T v W (5.2.6)
1
33
3
mp
1 1 4 4
fs 2W 2 3000
3 3 3
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