Chapter 3: Pulse Modulation
CHAPTER 3
PULSE MODULATION
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
3.7 Pulse Code Modulation
3.8 Noise in PCM Systems
3.9 Time Division Multiplexing
3.10 Digital Multiplexers
3.11 Modifications of PCM
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Chapter 3: Pulse Modulation
3.7 Pulse Code Modulation
• This part deals with the most basic form of
digital modulation.
• It is based on the two main processes we
have studied - the sampling process and the
quantization process.
• Definition: Pulse Code Modulation is a
technique where the message signal is
represented by a sequence of coded pulses.
It realizes digital representation of the
signal both time-wise and amplitude-wise.
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Chapter 3: Pulse Modulation
PCM
– essentially an analog-to-digital conversion (delta modulation (DM) and
differential pulse code modulation (DPCM));
– special – information contained in the instantaneous sample is
represented by digital words in a serial bit stream.
Transmitter
– sampling
– quantization (A/DC)
– encoding (A/DC)
Receiver
– regeneration
– decoding
– reconstruction
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Chapter 3: Pulse Modulation
The basic elements of a PCM system.
Figure 3.13
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Chapter 3: Pulse Modulation
PCM Transmission System
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Chapter 3: Pulse Modulation
Sampling
•
•
•
•
train of narrow rectangular pulses
> 2W (sampling theorem)
low-pass filter – anti-aliasing effect
result = limited number of discrete values per
second
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Chapter 3: Pulse Modulation
Quantization
• uniform law (described in sec.3.6)
• non-uniform – (voice applications); step size
increases in accordance with input-output
amplitude separation from origin
– compressor + uniform quantizer
– µ-law (m and v – normalized I/O voltages)
log(1   | m |)
| v |
log(1   )
(3.48)
– µ-law - |m| >>1 – logarithmic; |m| << 1 – linear
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Chapter 3: Pulse Modulation
Compression laws. (a)  -law. (b) A-law.
Figure 3.14
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Chapter 3: Pulse Modulation
A-law
1
 A| m|
,
0

|
m
|

 1  log A
A

| v | 
1  log( A | m |)
1
,
| m | 1

 1  log A
A
(3.50)
1  log A
,

A
d |m| 

d |v| 
 (1  A) | m |,

0 | m |
1
A
1
| m | 1
A
(3.51)
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Chapter 3: Pulse Modulation
Transmission side - Encoding
• Aim – robust to noise, interference and
channel impairments (see Table 3.2/204)
– line codes
– differential codes
• discrete set of values – appropriate signal
• binary codes – 1 and 0 (resistant to high noise
ratio) – 256 q. levels – 8 bit code word
• ternary codes - 1, 0 and -1
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Chapter 3: Pulse Modulation
Line codes for the electrical
representations of binary data.
(a) Unipolar non-return-to-zero
(NRZ) signaling (on-off
signalling).
(b) Polar NRZ signaling.
(c) Unipolar return-to-zero (RZ)
signaling.
(d) Bipolar RZ signaling.
(e) Split-phase or Manchester
code.
Figure 3.15
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Chapter 3: Pulse Modulation
Bandwidth of PCM Signals
What is the spectrum of a PCM data waveform
– For PAM – obtained as a function of the spectrum of the input analog
signal, because PAM is a linear function of the signal
– PCM is non-linear function of the input analog signal
– Spectrum is not directly related to the spectrum of the input analog
signal
Bandwidth depends on: bit rate and pulse shape used to
represent the data
– R  nf s
where n is the number of bits in the PCM word, f s sampling
frequency. For no aliasing, f s  2B . (B is the analog signal bandwidth).
1
1
– Dimensionality theorem gives the bounds:
B
PCM
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2
R
2
nf s
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Chapter 3: Pulse Modulation
Bandwidth of PCM Signals
1
1
R  nf s
2
2
• Min bandwidth
is for the case of (sin x) / x .
• Exact bandwidth depends on the type of line
encoding used (unipolar NRZ, polar NRZ, bipolar RZ
etc.
• Next slides provide information of bandwidth and
power requirements for different line encoding
schemes.
• For rectangular pulses first null bandwidth is:
B
PCM
 R  nf s so lower bound for PCM is B PCM  nB.
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Chapter 3: Pulse Modulation
Bandwidth of PCM Signals
• Finally, bandwidth for PCM signals in the case
where sampling is higher than fs , is
significantly higher than the corresponding
analog signal it represents.
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Chapter 3: Pulse Modulation
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Chapter 3: Pulse Modulation
Power spectra of line
codes:
Assumptions:
2. Average power is
normalized to unity
1. Symbols 1 and 0 are
equiprobable
3. Frequency is normalized
to the bit rate 1/Tb
Figure 3.16a
(a) Unipolar NRZ signal.
Disadvantages – DC component; power spectra – not 0 at
0 freq.
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Chapter 3: Pulse Modulation
Average power is
normalized to unity
Figure 3.16b
Frequency is normalized
to the bit rate 1/Tb
(b) Polar NRZ signal.
Disadvantages – large power near zero frequency
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Chapter 3: Pulse Modulation
Figure 3.16c
(c) Unipolar RZ signal.
Advantages – presence of delta function at f=0, 1/Tb- used
for sync
Disadvantage – 3dB more power polar RZ for same error
probability
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Chapter 3: Pulse Modulation
Figure 3.16d
(d) Bipolar RZ signal.
Advantages – no DC component; bipolar AMI
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Chapter 3: Pulse Modulation
Figure 3.16e
(e) Manchester-encoded signal.
Advantages – no DC; insignificant low-frequency
components
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Chapter 3: Pulse Modulation
Differential Codes
• encoding based on signal transitions
• reference signal (1) is necessary
Figure 3.17
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Chapter 3: Pulse Modulation
Transmission Path - Regeneration
• PCM advantage – control effects of noise and
distortion
• PCM signal – reconstructed by a series of
regenerative repeaters along the transmission route
• functions:
– equalization – reshaping, compensates for noise and
distortion
– timing – circuitry to provide a periodic pulse train for
determining sampling instants
– decision making – comparison to a predetermined threshold
Note: Occasional wrong decisions = bit errors
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Chapter 3: Pulse Modulation
Regeneration
• Possible problems:
– Noise and interference on the channel can add
resulting in wrong decisions = bit errors
– Spacing between pulses can deviate from
originally assigned = jitter
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Chapter 3: Pulse Modulation
Block diagram of regenerative repeater.
Figure 3.18
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Chapter 3: Pulse Modulation
Receiving side - Decoding
• Receiver side functions
– regeneration
– regrouping into code-words
– decoding
• Decoding: generating a pulse the amplitude of
which is the linear sum of all pulses in the
code word, with each pulse being weighted by
its place value (20, 21,…2R-1)
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Chapter 3: Pulse Modulation
Filtering
• Final operation – after decoder low-pass
reconstruction filter with bandwidth W
(message bandwidth).
• If transmission path is error free the recovered
signal has:
– no noise from channel
– only distortion - quantization
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
3.7 Pulse Code Modulation
3.8 Noise in PCM Systems
3.9 Time Division Multiplexing
3.10 Digital Multiplexers
3.11 Modifications of PCM
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Chapter 3: Pulse Modulation
3.8. Noise Considerations in PCM
Systems
• Two major sources:
– channel noise
– quantization noise – signal dependent
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Chapter 3: Pulse Modulation
Channel and Quantization Noise
• Channel Noise
– Introduces bit errors
– Fidelity – average probability of symbol errors (probability
that the reconstructed symbol differ from the transmitted
binary symbol); in BER (equal or weighted).
– Modeling - AWGN; reduce distance between repeaters;
performance dependent on quantization noise
• Quantization noise –presented before; design stage
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Chapter 3: Pulse Modulation
Error Threshold
• BER due to AWGN depends on Eb/N0 – ratio
of the transmitted signal energy per bit Eb, to
the noise spectral density N0.
• Table 3.3 – different behavior below and
above 11 dB. (compare to - 60-70 dB for high
quality speech transmission with AM).
• No error accumulation – regeneration
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
3.7 Pulse Code Modulation
3.8 Noise in PCM Systems
3.9 Time Division Multiplexing
3.10 Digital Multiplexers
3.11 Modifications of PCM
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Chapter 3: Pulse Modulation
3.9. Time Division Multiplexing
Figure 3.19
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Chapter 3: Pulse Modulation
Concept
• 1. Restricting each input by low-pass anti-aliasing
filter
• 2. Commutator – takes sample from each input
message (f > 2W); interleave samples in a frame Ts;
• 3. Pulse modulator – transformation for transmission
over common channel
• 4. Pulse demodulator
• 5. Decommutator – synchronized with the
commutator
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Chapter 3: Pulse Modulation
Synchronization
• TDM - Easy to add and drop sources
• Pulses duration considerations
– time interval limited by the sampling rate (reciprocal)
– more users – shorter pulses – difficult to generate; highly
influenced by impairments
– upper limit of number of independent sources
• Transmitter-receiver clock sync – very important –
two local clocks
– separate code element or pulse at the end of a frame
– orderly procedure for detecting sync pulses – searching
procedure
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Chapter 3: Pulse Modulation
Example: The T1 System
• 24 voice channels; separate pairs of wires; regeneration every 2 km; basic
to the North American Digital Switching Hierarchy
• Voice signal (300 – 3100 Hz) – low pass filter (cutoff frequency 3.1 kHz) –
Nyquist sampling rate = 6.2 kHz – actual sampling rate 8 kHz
• Companding - µ-law; µ = 255; 15 piece linear segment for approximating
the logarithmic characteristic; 1a, 2a, 3a … segments above x, 1b, 2b,
3b,…below x; 14 segments, each segment contains 16 uniform decision
levels
• for segment 0 – quantizer inputs are: ±1,±3, …±31 and the outputs are 0,
±1, ….±15; for segment 1a and 1b the decision level quantizer inputs are:
±31, ±35, …±95 and the outputs are ±16, ±17,…±31 and so on for the
other linear segments (up to 7a and 7b).
• Finally we have equally spacing on the y axis corresponding to non-equally
spaced inputs on the x axis (different step for different segment);
• Total representation levels: 31 + 14X16 = 255 for the 15 segment
companding characteristic;
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Chapter 3: Pulse Modulation
• Each of the 24 voice channels uses binary code with
8-bit word.
– first bit – 1 (positive voice input), 0 (negative voice input)
– bits 2 – 4 – identify particular segment
– last 4 bits – actual representation level (16 levels)
• Frames
– for 8 kHz, each frame occupies a period of 125 µs
– contains 24 X 8 =192 bit words; 1 bit for sync = 193 bits
– bit duration = 0.647 µs (125µs/193bits); transmission rate
1.544 Mb/s
• Signaling – every 6th frame, last bit; signaling rate for
each channel - 8 kHz/6 = 1.333 kb/s
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
3.7 Pulse Code Modulation
3.8 Noise in PCM Systems
3.9 Time Division Multiplexing
3.10 Digital Multiplexers
3.11 Modifications of PCM
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Chapter 3: Pulse Modulation
3.10. Digital Multiplexers
Same concept (TDM) used for multiplexing digital signals of
different rates.
Conceptual diagram of multiplexing-demultiplexing.
Figure 3.20
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Chapter 3: Pulse Modulation
• Multiplexing is accomplished by bit-by-bit
interleaving; selector switch – sequentially
scanning incoming line; at the receiving side –
separation into low speed components.
• Types of multiplexers:
– relatively low data bit rate user streams are TD
multiplexed over the public switched telephone
network.
– data transmission service by telecommunication
carriers; part of the national digital TDM
hierarchy.
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Chapter 3: Pulse Modulation
North American Digital TDM Hierarchy
• First level multiplexers – 24 64 kb/s streams (primary
rate) into a DS1 (digital signal 1) stream of 1.544
Mb/s carried on the T1 system.
• Second level multiplexers – 4 DS1 streams into a
DS2 stream at 6.312 Mb/s
• Third level multiplexers – 7 DS2 streams into a DS3
stream at 44.736 Mb/s
• Fourth level multiplexers – 6 DS3 into a DS4 stream
at 274.176 Mb/s
• Fifth level multiplexers – 2 DS4 streams into a DS5
at 560.160 Mb/s
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Chapter 3: Pulse Modulation
Important Note:
• Digital transmission facilities ONLY carry bit
streams without interpreting what the bits
themselves mean.
• The two sides have common understanding of
how to interpret the bits: voice, data, framing
format, signaling format etc.
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Chapter 3: Pulse Modulation
Problems:
• 1. Digital signals cannot be directly interleaved
into a format that allows for their separation
automatically. Common clock or perfect
synchronizations is needed.
• The multiplexed signal must include some
form of framing so the individual streams can
be identified at the source.
• The multiplexer should be able to handle small
variations in bit rates – bit stuffing.
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Chapter 3: Pulse Modulation
Bit stuffing
• To make the outgoing rate of the multiplexer a
little bit higher than the sum of the max
expected input rates.
• Each input is fed into an elastic store at the
multiplexer (reading can be done at different
rate).
• Identify stuffed bits – example AT&T M12
Multiplexer.
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Chapter 3: Pulse Modulation
Example: signal format of the AT&T
M12 Multiplexer
• Designed to combine 4 DS1 into one DS2 bit stream
• Each frame contains total of 24 control bits, separated
by sequences of 48 data bits
• 4 frames, transmitted one after the other
• 12 bits from each input bit-by-bit interleaved, 48 bits
• Four types of control bits – F,M and C inserted by
multiplexer – total of 24 control bits
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Chapter 3: Pulse Modulation
Signal format of AT&T M12 multiplexer
Figure 3.21
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Chapter 3: Pulse Modulation
Control bits
• F – 2 per subframe; main framing pulses (01010101)
• M – 1 pr subframe; secondary framing, identifying
the subframes (0111)
• C – 3 per subframe; stuffing indicators; indexes
denote input channel;
– first subframe has 3 C bits, indicating stuffing in first DS1
stream; value 1 of all three indicates stuffed bits; value 0 –
no stuffed bits; majority logic decoding
– if there is stuffing position of stuffing is – first bit after F1
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Chapter 3: Pulse Modulation
Receiver
• 1. Searches for main framing sequence – 01010101 in
F bits
• 2. Establishes the identity of the four DS1 streams
and position of M and C bits
• 3. From the position of the M bits the correct position
of the C bits is verified
• 4. Streams properly demultiplexed and destuffed.
• Safeguards:
– Double checking F and M bits for framing.
– Single error correction capability built into the C-control
bits ensures that the 4 DS1 streams are properly destuffed
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Chapter 3: Pulse Modulation
Outline
•
•
•
•
•
3.7 Pulse Code Modulation
3.8 Noise in PCM Systems
3.9 Time Division Multiplexing
3.10 Digital Multiplexers
3.11 Modifications of PCM
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Chapter 3: Pulse Modulation
3.11. Virtues, Limitations and
Modifications of PCM
•
•
•
•
Advantages:
1. Robustness to channel noise and interference.
2. Signal regeneration possibilities along the path.
3. Efficient trade-off between increased bandwidth
and improved SNR (exponential law)
• 4. Integration of different base-band signals.
• 5. Comparative easy of add and drop sources.
• 6. Secure communication (special modulation,
encryption).
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Chapter 3: Pulse Modulation
• Disadvantages:
• 1. Increases complexity - VLSI technology
• 2. Increased bandwidth – satellites and fiber
optic cables; data compression techniques;
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Chapter 3: Pulse Modulation
Home reading assignment
• Conditions for optimality of Scalar Quantizers
–Haykin, p.198 – 201.
• Provide one A4 page summary on what you
have read. To be uploaded on the site.
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