Sampling
PAM- Pulse Amplitude Modulation
(continued)
EELE445-14
Lecture 14
Sampling
Properties will be looking at for:
•Impulse Sampling
•Natural Sampling
•Rectangular Sampling
1
Figure 2–18 Impulse sampling.
Impulse Sampling
2
Ts
δ T (t ) =
s
ϕn = 0
δ T (t ) =
s
∞
∞
∑ δ (t − nT ) ⇒ D + ∑ D
s
n
0
n = −∞
ωs =
cos(nωs t + ϕ n )
n =1
2π
Ts
D0 =
1
Ts
Dn =
2
Ts
1
[1 + 2 cos(ωst ) + 2 cos(2ωst ) + 2 cos(3ωst ) + L]
Ts
2
Impulse Sampling
ws (t )
δ T (t ) =
s
∞
∑ δ (t − nT )
n = −∞
s
ws (t ) = w(t )δ Ts (t ) =
w(t )
[1 + 2 cos(ωs t ) + 2 cos( 2ωs t ) + 2 cos( 3ωs t ) + L]
Ts
Ws ( f ) = F [ws (t )] =
1
Ts
∞
∑W ( f − nf
n = −∞
s
)
Impulse Sampling- text
ws (t ) = w(t )∑n = −∞ δ (t − nTs )
∞
= ∑n = −∞ w( nTs )δ (t − nTs )
∞
eq 2 − 171
subsituting ,
ws (t ) = w(t )∑n = −∞
∞
Ws ( f ) =
1
Ts
∑
∞
n = −∞
1 jnωst
e
Ts
W ( f − nf s )
3
Impulse Sampling
The spectrum of the impuse sampled signal is the spectrum of the
unsampled signal that is repeated every fs Hz, where fs is the
sampling frequency or rate (samples/sec). This is one of the
basic principles of digital signal processing.
Note:
This technique of impulse sampling is often used to
translate the spectrum of a signal to another frequency band that
is centered on a harmonic of the sampling frequency, fs.
If fs>=2B, (see fig 2-18), the replicated spectra around
each harmonic of fs do not overlap, and the original spectrum can
be regenerated with an ideal LPF with a cutoff of fs/2.
Impulse Sampling
Undersampling and aliasing.
4
Natural Sampling
Generation of PAM with natural sampling (gating).
Natural Sampling
Duty cycle =1/3
5
Natural Sampling
Null at
fs
= 3 fs
d
PAM and PCM
• PAM- Pulse Amplitude Modulation:
– The pulse may take any real voltage value that is
proportional to the value of the original waveform.
No information is lost, but the energy is
redistributed in the frequency domain.
• PCM- Pulse Code Modulation:
– The original waveform amplitude is quantized with
a resulting loss of information
6
Figure 3–4 Demodulation of a PAM signal (naturally sampled).
nth Nyquist region recovery
Comb Oscillator
n=1,2,3….
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©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–5 PAM signal with flat-top sampling.
Impulse sample and hold
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©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
7
Figure 3–6 Spectrum of a PAM waveform with flat-top sampling.
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–7 PCM trasmission system.
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
8
PCM Transmission
• Negative: The transmission bandwidth
of the PCM signal is much larger than
the bandwidth of the original signal
• Positive: The transmission range of a
PCM signal may be extended with the
use of a regenerative repeater.
Figure 3–8 Illustration of waveforms in a PCM system.
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©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
9
Figure 3–8 Illustration of waveforms in a PCM system.
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Illustration of waveforms in a PCM system.
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
10
Illustration of waveforms in a PCM system.
•The error signal, (quantization noise), is inversely
proportional to the number of quantization levels used
to approximate the waveform x(t)
•Companding is used to improve the SQNR (signal to
quantization noise ratio) for small amplitude x(T) signals
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–9 Compression characteristics (first quadrant shown).
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
11
Figure 3–9 Compression characteristics (first quadrant shown).
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–10 Output SNR of 8-bit PCM systems with and without companding.
⎛ V peak range ⎞
⎡S⎤
⎟⎟
⎢⎣ N ⎥⎦ = 4.77 + 6.02n − 20 log⎜⎜ x
dB
rms
⎝
⎠
uniform quantion
⎡S⎤
⎢⎣ N ⎥⎦ = 4.77 + 6.02n − 20 log[ln (1 + μ )] μ − law companding
dB
⎡S⎤
⎢⎣ N ⎥⎦ = 4.77 + 6.02n − 20 log[1 + ln ( A)] A − law companding
dB
Couch, Digital and Analog Communication Systems, Seventh Edition
©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
12
Quantization Noise,
Analog to Digital Converter-A/D
EELE445
Lecture 16
26
Figure 3–7 PCM trasmission system.
q(x)
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©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
13
Quantization
xˆn = Q ( x )
V
x(t)
Δ
Δ=
2V
M
=
V
2 n −1
M = 2n
Q(x)
-V
Quantization – Results in a Loss of Information
x(t)
Δ=
Δ
2
Q(x)
−
Δ
2
2V
M
=
V
2n −1
Lost Information
After sampling, x(t)=xi
xi ∈ R
After Quantization:
Q(x) = x̂, x ∈ R
14
Quantization Noise
Quantization function:
Define the mean square distortion:
q( x ) = ( x − Q ( x )) 2 = ~
x2
and
Δ
x − Q( x) ≤
2
q2 =
Δ
2
1
q 2 dq
∫
Δ −Δ
Quantization
Δ=
2V
M
=
V
2 n −1
2
M =2
Δ2
=
12
2
(
V)
=
= Pnq the quantization noise
3M 2
n
where M=2n, V is ½ the A/D input range,
and n is the number of bits
15
Quantization Noise from the expectation operator:
~
Since X is a random variable, so are Xˆ and X
So we can define the mean squared error (distortion) as :
D = E d X , Xˆ = E ( X − Q ( X )) 2
[(
)] [
]
~
The pdf of the error is uniformly distributed : X = X − Q ( X )
f (x~ )
1
Δ
−
Δ
2
Δ
2
⎧Δ
⎪
f (~
x) = ⎨ 2
⎪⎩0
~
x
Δ ~ Δ
≤x≤
2
2
otherwise
−
SQNR – Signal to Quantization Noise
Ratio
16
SQNR – Signal to Quantization Noise
Ratio
Px may be found using:
SQNR – Signal to Quantization Noise Ratio
The distortion, or “noise”, is therefore:
Where Px is the power of the input signal
17
SQNR – Linear Quantization
[ ]
2
E X 2 ≤ xmax
Px
<1
2
xmax
The SQNR decreases as
The input dynamic range
increases
U-Law Nonuniform PCM
used to increase SQNR for given Px , xmax, and n
U=255 U.S
18
A-Law Nonuniform PCM
a=87.56 U.S
u-Law v.s. Linear Quantization
8 bit
Px is signal power
Relative to full scale
Px
19
Pulse Code Modulation, PCM, Advantage
compared with analog systems
• b is the number of bits
• γ is (S/N)baseband
Relative to full scale
• PPM is pulse position
modulation
20