Sampling Theorem Quantization Noise and its types Encoding-PCM Power of Signal Signal to noise Ratio Channel Capacity Nyquist Bandwidth Shannon Capacity Formula Multirate Digital Signal Processing-Up/Down Sampling Applications

1. Topics
•Sampling Theorem
•Quantization
•Noise and its types
•Encoding-PCM
•Power of Signal
•Signal to noise Ratio
•Channel Capacity
•Nyquist Bandwidth
•Shannon Capacity Formula
•Multirate Digital Signal Processing-
Up/Down Sampling
•Applications

2. Analog-to-Digital Conversion
Quantize r
Sampler Coder
x(t)
2
0101...
X(n)
Discrete-
time
signal
x (t)
q
Quantiz
ed
Signal
DigitalSignal
Sampling:
conversion from cts-time to dst-time by taking samples" at discrete time instants
uniform sampling: x(n) = xa(nT) where T is the sampling period and n ε Z
E.g.,
Quantization: conversion from dst-time cts-valued signal to a dst-
time dst-valued signal quantization error: eq(n) = xq(n)- x(n))
Coding:representation of each dst-value xq(n) by a b-bit binary sequence

3. Sampling Theorem
If the highest frequency contained in an analog signal xa(t) is Fmax = B and the signal is
sampled at a rate
Fs > 2Fmax=2B
then xa(t) can be exactly recovered from its sample values using the interpolation
function
Therefore, given the interpolation relation, xa(t) can be written as
where xa(nT) = x(n); called band limited interpolation.
3
Note: FN = 2B = 2Fmax
is called the Nyquist rate

4. 4

5. 5

7. 7

8. 8

9. Noise (1)
• Additional signals inserted between transmitter
and receiver
1. Thermal
—Due to thermal agitation of electrons
—Uniformly distributed
—White noise, cannot be eliminated
2. Intermodulation
—Signals that are the sum and
difference of original frequencies
sharing a medium
Intermodulation
9

10. Noise (2)
3. Crosstalk
—A signal from one line is picked up by another
4. Impulse
—Irregular pulses or spikes
• e.g. External electromagnetic interference
—Short duration
—High amplitude
—Sharp spike could change a 1 to 0 or
a 0 to 1. 10

11. Thermal Noise
11
• The amount of thermal,noise to e found in a
bandiwdth of 1 Hz in any device or conductor is:
• N0 = kT (W/Hz)
• N0 = noise power density in watt per 1 Hz of bandwidth
• k = boltzman constant = 1.38 x 10
• T = temp, in Kelvins
• Thermal noise in watt present in a bandwidth of B
• N = kTB = 10 log k + 10 log T + 10 log B

12. Signal to noise ratio
:It is either unit-less or specified in dB. The S/N ratio may be specified
anywhere within a system.

13. Channel Capacity
Def. :Max. rate at which data can be transmitted
over a given com. path/channel under given
condition
Concept of channel capacity:
• Data rate
—In bits per second
—Rate at which data can be communicated
• Bandwidth
—Range of frequency
—In cycles per second or Hertz,
(unit for frequency, f= 1/T)
Data a
n
d
—C
o
Cm
op
u
nt
e
srtrained by transmitter and medium 13

14. Channel Capacity (cont)
• Noise
—Average level of noise over com. path
• Error rate
—Rate at which errors occur
—Reception of a 1 when 0 was transmitted or the
other way
• Com. facilities are expensive
—Bandwidth cost
—Make efficient use of given bandwidth
—Main constraint in achieving this efficiency is noise
14

15. Nyquist Bandwidth
• In the case of a channel that is noise free,
limitation on data rate is simply the bandwidth
of the signal.
• If rate of signal transmission is 2B then signal
with frequencies no greater than B is sufficient
to carry signal rate
—Given bandwidth B, highest signal rate is 2B
—Given binary signal, data rate supported by B Hz is
2B bps
• Can be increased by using M signal levels
C= 2B log2M
C = Channel capacity 31

16. Nyquist Bandwidth (2)
16
• Can be increased by using M signal levels
C= 2B log2M
• M is the number of discrete signal or voltage
levels
• Eg: 1 bit value (0 to 1) = 21 (M= 2 levels)
2 bit value (00 to 11) =22 (M= 4 levels)
3 bit value (000 to 111) =23 (M= 8 levels)

17. Shannon Capacity Formula
17
• Nyquist’s formula indicates that doubling the
bandwidth doubles the data rate.
• Consider data rate, noise and error rate
• Faster data rate shortens each bit, so burst of
noise affects more bits
—At given noise level, high data rate means higher
error rate
—Greater signal strength would improve ability to
receive data correctly in the presence of noise.

18. Shannon Capacity Formula (2)
• Formula developed by a mathematician Claude
Shannon.
• Signal to noise ratio (in decibels)
SNRdb 10 log10 (signal/noise)
=
• Ex.: Suppose that Vs = 10.0 μv
and Vn = 1.00 μv . Then
S/N = 20 log(10(1.00)) = 20.0 dB
• Max. channel capacity
C=B log2(1+SNR)
•a
t
a
Ta
n
hdiC
so
m
ip
su
t
e
er rror free capacity 18

19. 19
Multirate Digital Signal
Processing
• systems that employ multiple sampling rates in
the processing of digital signals are called
multirate digital signal processing systems.
• Multirate systems are sometimes used for
sampling-rate conversion
• In most applications multirate systems are
used to improve the performance, or for
increased computational efficiency.

20. Multirate Digital Signal
Processing
• The basic Sampling operations in a
multirate system are:
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21. Sampling Rate Reduction by Integer Factor D
21

22. Sampling Rate Reduction by Integer
Factor D
H(n)
Downsampler
D
Digital anti-aliasing
filter
22
Sampling-rate
compressor

23. Sampling Rate Increase by
Integer Factor I
• Interpolation by a factor of L, where L is a
positive integer, can be realized as a two-step
process of upsampling followed by an anti-
imaging filtering.
L LPF
23

24. Sampling Rate Increase by Integer
Factor I
• An upsampling operation to a discrete-time
signal x(n) produces an upsampled signal
y(m) according to
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25. 25
Applications of
Multirate DSP
• Multirate systems are used in a CD player when the
music signal is converted from digital into analog
(DAC).

26. Applications of
Multirate DSP
26

27. Applications of
Multirate DSP
The effect of oversampling also has someother
desirable features:
 Firstly, it causes the image frequencies to be much
higher and therefore easier to filter out.
 Secondly reducing the noise power spectral
density, by spreading the noise power over a
larger bandwidth.
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