Chapter 3: Delta Modulation
CHAPTER 3
DELTA MODULATION
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Chapter 3: Delta Modulation
Outline
• 3.12 Delta Modulation
Delta Sigma Modulation
• 3.13 Linear Prediction
• 3.14 Differential Pulse Code Modulation
• 3.15 Adaptive Differential Pulse Code
Modulation
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Chapter 3: Delta Modulation
3.12 Delta Modulation
• Definition: Delta Modulation is a technique
which provides a staircase approximation to
an over-sampled version of the message
signal (analog input).
• sampling is at a rate higher than the Nyquist
rate – aims at increasing the correlation
between adjacent samples; simplifies
quantizing of the encoded signal
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Chapter 3: Delta Modulation
Illustration of the delta modulation process
Figure 3.22
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Chapter 3: Delta Modulation
Principle Operation
• message signal is over-sampled
• difference between the input and the
approximation is quantized in two levels - +/-Δ
• these levels correspond to positive/negative
differences
• provided signal does not change very rapidly
the approximation remains within +/-Δ
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Chapter 3: Delta Modulation
Assumptions and model
We assume that:
• m(t) denotes the input message signal
• mq(t) denotes the staircase approximation
• m[n] = m(nTs), n = +/-1, +/-2 … denotes a
sample of the signal m(t) at time t=nTs, where
TS is the sampling period
• then
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Chapter 3: Delta Modulation
• we can express the basic principles of the delta
modulation in a mathematical form as follow:
e[n]  m[n]  mq [n 1]
error signal
eq   sgn(e[n])
(3.52)
(3.53)
quantized
error signal
mq [n]  mq [n 1]  eq [n]
quantized
output
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Chapter 3: Delta Modulation
Pros and cons
• Main advantage – simplicity
• Sampled version of the message is applied to a
modulator (comparator, quantizer,
accumulator)
• delay in accumulator is “unit delay” = one
sample period (z-1)
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Chapter 3: Delta Modulation
Figure 3.23
DM system.
(a) Transmitter.
(b) Receiver.
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Chapter 3: Delta Modulation
Transmitter Side
• comparator – computes difference between input signal and
one interval delayed version of it
n
• quantizer – includes a hardmq [n]  
sgn(e[i ])
limiter with an input-output
relation a scaled version of
i 1
the signum function
n
• accumulator – produces the

eq [i ]
approximation mq[n] (final
i 1
result) at each step by
adding either +Δ or –Δ
• = tracking input samples by
(3.55)
one step at a time


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Chapter 3: Delta Modulation
Receiver Side
• decoder – creates the sequence of positive or
negative pulses
• accumulator – creates the staircase
approximation mq[n] similar to tx side
• out-of-band noise is cut off by low-pass filter
(bandwidth equal to original message
bandwidth)
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Chapter 3: Delta Modulation
Noise in Delta Modulation Systems
• slope overhead distortion
• granular noise
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Chapter 3: Delta Modulation
Slope Overhead Distortion
• The quantized message
signal can be
mq [n]  m[n]  q[n]
(3.56)
represented as:
• where the input to the
e[n]  m[n]  m[n  1]  q[n  1]
quantizer can be
(3.57)
represented as:
Sample of m(T)
at time nT
Quantizer input
at time (n-1)T
So, (except for the quantization error) the quantizer input
is the first backward difference (derivative) of the input
signal = inverse of the digital integration process
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Chapter 3: Delta Modulation
Discussion
• Consider the max slope
of the input signal m(t)
• To increase the samples
{mq[n]} as fast as the
input signal in its max
slope region the
following condition
should be fulfilled:

dm(t )
 max
Ts
dt
(3.58)
otherwise the
step-size Δ is too
small
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Chapter 3: Delta Modulation
Granular Noise
•
•
•
•
In contrast to slope overhead
Occurs when step size is too large
Usually relatively flat segment of the signal
Analogous to quantization noise in PCM systems
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Chapter 3: Delta Modulation
Conclusion:
• 1. Large step-size is necessary to
accommodate a wide dynamic range
• 2. Small step-size is required for accuracy with
low-level signals
• = compromise between slope overhead and
granular noise
• = adaptive delta modulation, where the step
size is made to vary with the input signal
(3.16)
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Chapter 3: Delta Modulation
Outline
• 3.12 Delta Modulation
Delta Sigma Modulation
• 3.13 Linear Prediction
• 3.14 Differential Pulse Code Modulation
• 3.15 Adaptive Differential Pulse Code
Modulation
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Chapter 3: Delta Modulation
Delta Sigma Modulation
• Conventional delta modulation - Quantizer
input is an approximation of the derivative of
the input message signal m(t).
• Results in the accumulation of error (noise)
– accumulated noise (transmission disturbances) at
the receiver (cumulative error).
• Possible solution: integrating the message
before delta modulation – called delta sigma
modulation
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Chapter 3: Delta Modulation
Remark 1:
• The message signal is defined in its continuous
form – so pulse modulator contains a hard
limiter and a pulse generator to produce a 1-bit
encoded signal
• integration at the tx requires differentiation at
the rx side.
• But: As in conventional DM the message has
to be integrated at the final stage this
eliminates the need of differentiation here.
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Chapter 3: Delta Modulation
Block diagrams of systems for
realizing Delta-Sigma Modulation
Figure 3.25
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Chapter 3: Delta Modulation
Remark 2:
• Integration is a linear operation
• Int 1 and Int 2 can be combined in a single
integrator placed after the comparator
(previous slide – 3.25 b)
• Results in a simpler version of DSM scheme
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Chapter 3: Delta Modulation
Pros and cons for DSM
• Simplicity of implementation both at the tx
and rx side
• Requires sampling rate far in excess of the
Nyquist rate (PCM) – increase in transmission
and channel bandwidth
• If bandwidth is at a premium we have to
choose increased system complexity
(additional signal processing) to achieve
reduced bandwidth.
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Chapter 3: Delta Modulation
How does it work?
• Reading assignment:
• 3.13 Linear Prediction
(plus all that you are taught in the Signals and
Systems – part II)
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Chapter 3: Delta Modulation
Outline
• 3.12 Delta Modulation
Delta Sigma Modulation
• 3.13 Linear Prediction
• 3.14 Differential Pulse Code Modulation
• 3.15 Adaptive Differential Pulse Code
Modulation
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Chapter 3: Delta Modulation
3.14 Differential PCM
• Sampling at higher then Nyquist rate creates
correlation between samples (good and bad)
• Difference between samples has small variance –
smaller than the variance of the signal itself
• Encoded signal contains redundant information
• Can be used to a positive end – remove redundancy
before encoding to get a more efficient signal to be
transmitted
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Chapter 3: Delta Modulation
How it works – the background
• We know the signal up to a certain time
• Use prediction to estimate future values
• Signal sampled at fs= 1/Ts ; sampled sequence –
{m[n]}, where samples are Ts seconds apart
• Input signal to the quantizer – difference between
the unquantized input signal m(t) and its prediction:
e[n]  m[n]  mˆ [n]
(3.74)
prediction of the input
sample
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Chapter 3: Delta Modulation
• Predicted value – achieved by linear prediction
filter whose input is the quantized version of
the input sample m[n].
• The difference e[n] is the prediction error
(what we expect and what actually happens)
• By encoding the quantizer output we actually
create a variation of PCM called differential
PCM (DPCM).
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Chapter 3: Delta Modulation
Figure 3.28
DPCM system.
(a) Transmitter.
(b) Receiver.
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Chapter 3: Delta Modulation
Details:
• Block scheme is very similar to DM
• quantizer input
ˆ [n]
e[n]  m[n]  m
(3.74)
• quantizer output may be expressed as:
eq [n]  e[n]  q[n]
(3.75)
• prediction filter output may be expressed as:
ˆ [n]  eq [n]
mq [n]  m
(3.76)
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Chapter 3: Delta Modulation
If we substitute 3.75 into 3.76 we get:
ˆ [n]  e[n]  q[n]
mq [n]  m
Quantized input of
the prediction filter -
(3.77)
sum is equal to input
sample
mq [n]  m[n]  q[n]
(3.78)
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Chapter 3: Delta Modulation
Details – cont’d
• mq[n] is the quantized version of the input sample m[n]
• so, irrespective of the properties of the prediction filter the
quantized sample mq[n] at the prediction filter input differs
from the original sample m[n] with the quantization error q[n].
• If the prediction filter is good, the variance of the prediction
error e[n] will be smaller than the variance of m[n]
• This means that if we make a very good prediction filter
(adjust the number of levels) it will be possible to produce a
quantization error with a smaller variance than if the
input sample m[n] is quantized directly as in standard PCM
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Chapter 3: Delta Modulation
Receiver side
• decoder – constructs the quantized error signal
• quantized version of the input is recovered by
using the same prediction filter as at the tx
• if there is no channel noise – encoded input to
the decoder is identical to the transmitter
output
• then the receiver output will be equal to mq[n]
(differs from m[n] by q[n] caused by
quantizing the prediction error e[n])
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Chapter 3: Delta Modulation
Comparison
• DPCM and DM
– DPCM includes DM as a special case
– Similarities
• subject to slope-overhead and quantization error
– Differences
• DM uses a 1-bit quantizer
• DM uses a single delay element (zero prediction order)
• DPCM and PCM
– both DM and DPCM use feedback while PCM does not
– all subject to quantization error
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Chapter 3: Delta Modulation
Processing Gain
• Output signal-to-noise
ratio (SNRO)
•
•
σM2 – variance of m[n]
σQ2 – variance of quantization error q[n]
• rewrite using variance of
the prediction error σE2
 E2
( SNR)Q  2
Q
signal-to-quantization noise ratio

(SNR)O 

  M2
( SNR)O   2
 E
(3.81)
processing gain
2
M
2
Q
   E2 
  2   G p ( SNR)Q
 Q 
(3.80)
 M2
Gp  2
E
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Chapter 3: Delta Modulation
• The processing gain Gp when greater than unity
represents the signal-to-noise ratio that is due to the
differential quantization scheme.
• For a given input message signal σM is fixed, so the
smaller the σE the greater the Gp.
• This is the design objective of the prediction filter
• For voice signals – optimal main advantage of DPCM
over PCM is b/n 4-11 dB
• Advantage expressed in terms of bit rate (bits)
– 1 bit =6 dB of quantization noise (Table 3.35, p 198)
– So for fixed SNR, sampling rate 8 kHz – DCPM provides
saving of 8-16 kb/s (1 -2 bits per sample) PCM
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Chapter 3: Delta Modulation
Outline
• 3.12 Delta Modulation
Delta Sigma Modulation
• 3.13 Linear Prediction
• 3.14 Differential Pulse Code Modulation
• 3.15 Adaptive Differential Pulse Code
Modulation
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Chapter 3: Delta Modulation
3.15 Adaptive Differential PCM
• PCM for speech coding of 64 kb/s requires high
channel bandwidth
• some applications (secure transmission over radio
channel – low capacity)
• requires speech coding at low bit rates but preserving
acceptable fidelity (not 64 kb/s PCM but 32, 16, 8
etc)
• possible using special coders that utilize statistical
characteristics of speech signals and properties of
hearing
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Chapter 3: Delta Modulation
Design Objectives
• 1. Remove redundancies from speech signals
• 2. Assign available bits to encode non-redundant
parts of speech signal in an efficient way
• Standard PCM is at 64 kb/s – can be reduced to 32,
16, 8 or even 4 kb/s
• Price = proportionally increased complexity
– For same speech quality but Half the bit rate Computational complexity is an order of magnitude higher
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Chapter 3: Delta Modulation
ADPCM principles
• Allows encoding of speech at 32 kb/s – requires 4 bits per
sample
• Uses adaptive quantization and adaptive prediction
– adaptive quantization – uses a time-varying step Δ[n]. The stepsize is varied to match the input signal σM2
[n]  ˆ M [n]
(3.83)
• φ is a constant; the other – estimate of the standard
deviation - has to be computed continuously
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Chapter 3: Delta Modulation
• Two possibilities:
– adaptive quantization with forward estimation
(AQF) – uses unquantized samples of the input
signal to derive forward estimates of σM[n];
requires a buffer to store samples for a certain
learning period; incurs delay (~ 16 ms for speech)
– adaptive quantization with backward estimation
(AQB) – uses samples of the quantizer output to
derive backwards estimates of σM[n]
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Chapter 3: Delta Modulation
• Adaptive prediction in ADPCM
– adaptive prediction with forward estimation
(APF); uses unqunatized samples of the input
signal to calculate prediction coefficients;
disadvantages similar to AQF
– adaptive prediction with backward estimation
(APB); uses samples of the quantizer output and
the prediction error to compute predictor
coefficients; logic for adaptive prediction –
algorithm for updating predictor coefficients
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Chapter 3: Delta Modulation
Adaptive quantization with backward estimation (AQB).
Figure 3.29
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Chapter 3: Delta Modulation
Adaptive prediction with backward estimation (APB).
Figure 3.30
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Chapter 3: Delta Modulation
Conclusion:
• PCM at 64 kb/s and ADPCM at 32 kb/s are
internationally accepted standards for voice
coding and decoding.
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