EC 2301 Digital communication

Unit I and II Question bank

UNIT I Digital communication system
2 marks

1.Draw block diagram of digital communication system.

Information source and formatter Source Channel Baseband processor
input transducer encoder encoder or bandpass
modulator

channel

Output signal and output deformatter Source Channel Baseband decoder

transducer decoder decoder and bandpass
demodulator

2.Define Formatter

* In Digital communication system, the input signal should be in digital form so

that digital signal processing techniques can be employed on the signals.

* The block which converts the electrical signals at the output of the transducer
into a sequence of digital signals is known as Formatter.

3. What is the need for Base Band Processor?

*In low-speed transmission, the channel encoded signal is generally not modulated.
The transmission takes place in baseband.

*However, for proper detection of the signal and to combat noise and interference
line coding , pulse shaping and special filters are employed in the receiver.

*All these are collectively called as baseband processor. This is the case in fixed
telephony and data storage systems.

4.Define spectral efficiency and BER.

Spectral Efficiency:

The number of bits transmitted per second for every hertz of bandwidth is called
spectral efficiency.
BER(Bit Error Rate):

The ratio of the error bits to the transmitted bits is called as Bit Error Rate.

BER = Error Bits

Transmitted Bits

5. State Dimensionality theorem.

The theorem states that, A real waveform can be completely specified by N

independent pieces of information.

where

N is given by

N =2BT0

N is the dimension of the waveform in signal space.

B is the bandwidth of the signal.
T0 is the time over which signal waveform is being described.

6.Define PSD and symbol rate

PSD:

*The Power Spectral Density (PSD) refers to the amount of power per unit of
frequency as function of frequency. It describes how the power of signal or time series
is distributed with frequency. By knowing PSD, the system bandwidth and frequency can
be calculated.

*The PSD power spectrum of the stationary process is given by,

Sx(f)= Rx(τ) exp(-j2πfτ) dτ

Symbol Rate:

*Symbol rate or the Baud rate is defined as the number of symbols transmitted
per second.
*The symbol rate of a digital signal represented by N points which are
transmitted over an interval of T 0 second is given by,

Rs = N
T0

B = Rs
2
*Half of the symbol rate is bandwidth.

8.What is GSOP?

*GSOP is Gram Schmidt Orthogonalisation Procedure.

*The procedure for obtaining the basis set from the original set is called GSOP.

*It is used to construct a set of orthogonal basis function.

Ψk (t) = Ψ’k (t)

√ Ek

Where Ψk (t) is the k basis function

√ Ek is the normalized energy of the signal sk (t)

9.Define signal space and basis function.

Signal space:

The complete set of all signals is called signal space.

Basis function:

*Collection of minimum number of functions necessary to represent a given signal

is called Basis function.

*Basis functions are independent.

*Basis functions are always orthogonal to each other.

10.Define dimension of signal space and basis set

dimension of signal space:

The minimum number of basis function is called dimension of signal space.

basis set:

Collection of the basis function is called Basis set.

11.Define signaling set

In digital communication system only a few logical levels of input signal are
supported. A particular signal waveform is transmitted for each of these levels. The set of
all the signals is called signaling set.

12.Define Kroneckar delta function

Basis function,

ψi (t)ψk (t) dt = kj δjk ; 0≤t≤T , j, k = 1,………..N

Where

k is a non-zero constant

δjk is the Kronecker delta function

δjk = 1 for j=k

0 otherwise

13.List the mathematical models of communication channel

There are three mathematical models of communication channel.

Additive Noise channel

Linear filter channel
Linear time variant filter channel

14.Explain in brief about channel classification

Channels can be classified into

Wired channels
Eg: copper cable
Optical fiber (few Gbps)
co-axial cable(few 100 Mbps)
Ethernet cable
Wireless channels
Eg: underwater ocean channel carrying acoustic wave
Free space carrying electromagnetic wave
Infrared wave(few THz)

15.How to improve the performance measure of digital communication system?

The performance measure of digital communication system can be improved by the

following ways:

Improving modulation and demodulation techniques.

Improving coding and decoding techniques.
Improving pulse shaping and filtering techniques.
Spectral efficiency and BER.
Improving the transmitted power.
16 marks
1.Explain the functional description of digital communication system in
detail.
ELEMENTS OF DIGITAL COMMUNICATION SYSTEMS: The figure 1.2 shows
the functional elements of a digital communication system.
Source of Information:
1. Analog Information Sources.
2. Digital Information Sources. Analog Information Sources → Microphone actuated by a
speech, TV Camera scanning a scene, continuous amplitude signals. Digital Information
Sources → These are teletype or the numerical output of computer which consists of a
sequence of discrete symbols or letters. An Analog information is transformed into a
discrete information through the process of sampling and quantizing.

Digital Communication System

Fig 1.2: Block Diagram of a Digital Communication System SOURCE ENCODER /

DECODER: The Source encoder ( or Source coder) converts the input i.e. symbol
sequence into a binary sequence of 0‟ s and 1‟ s by assigning code words to the symbols
in the input sequence.
For eg. :-If a source set is having hundred symbols, then the number of bits used to
represent each symbol will be 7 because 27=128 unique combinations are available. The
important parameters of a source encoder are block size, code word lengths, average
data rate and the efficiency of the coder (i.e. actual output data rate compared to the
minimum achievable rate) At the receiver, the source decoder converts the binary output
of the channel decoder into a symbol sequence. The decoder for a system using fixed –
length code words is quite simple, but the decoder for a system using variable – length
code words will be very complex. Aim of the source coding is to remove the redundancy
in the transmitting information, so that bandwidth required for transmission is minimized.
Based on the probability of the symbol code word is assigned. Higher the probability,
shorter is the codeword. Ex: Huffman coding.

CHANNEL ENCODER / DECODER: Error control is accomplished by the channel

coding operation that consists of systematically adding extra bits to the output of the
source coder. These extra bits do not convey any information but helps the receiver to
detect and / or correct some of the errors in the information bearing bits. There are two
methods of channel coding:
1. Block Coding: The encoder takes a block of „k‟ information bits from the source
encoder and adds „r‟ error control bits, where „r‟ is dependent on „k‟ and error control
capabilities desired.
2. Convolution Coding: The information bearing message stream is encoded in a
continuous fashion by continuously interleaving information bits and error control bits.

The Channel decoder recovers the information bearing bits from the coded binary stream.
Error detection and possible correction is also performed by the channel decoder. The
important parameters of coder / decoder are: Method of coding, efficiency, error control
capabilities and complexity of the circuit.
MODULATOR: The Modulator converts the input bit stream into an electrical waveform
suitable for transmission over the communication channel. Modulator can be effectively
used to minimize the effects of channel noise, to match the frequency spectrum of
transmitted signal with channel characteristics, to provide the capability to multiplex
many signals.
DEMODULATOR: The extraction of the message from the information bearing
waveform produced by the modulation is accomplished by the demodulator. The output
of the demodulator is bit stream. The important parameter is the method of demodulation.

CHANNEL: The Channel provides the electrical connection between the source and
destination. The different channels are: Pair of wires, Coaxial cable, Optical fibre, Radio
channel, Satellite channel or combination of any of these. The communication channels
have only finite Bandwidth, non-ideal frequency response, the signal often suffers
amplitude and phase distortion as it travels over the channel. Also, the signal power
decreases due to the attenuation of the channel. The signal is corrupted by unwanted,
unpredictable electrical signals referred to as noise. The important parameters of the
channel are Signal to Noise power Ratio (SNR), usable bandwidth, amplitude and phase
response and the statistical properties of noise.

Advantages of Digital Communication

1. The effect of distortion, noise and interference is less in a digital communication
system. This is because the disturbance must be large enough to change the pulse
from one state to the other.
2. Regenerative repeaters can be used at fixed distance along the link, to identify and
regenerate a pulse before it is degraded to an ambiguous state.

3. Digital circuits are more reliable and cheaper compared to analog circuits.

4. The Hardware implementation is more flexible than analog hardware because of the
use of microprocessors, VLSI chips etc.

5. Signal processing functions like encryption, compression can be employed to maintain

the secrecy of the information.

6. Error detecting and Error correcting codes improve the system performance by
reducing the probability of error.

7. Combining digital signals using TDM is simpler than combining analog signals using
FDM. The different types of signals such as data, telephone, TV can be treated as
identical signals in transmission and switching in a digital communication system.

8. We can avoid signal jamming using spread spectrum technique.

Disadvantages of Digital Communication:

1. Large System Bandwidth:- Digital transmission requires a large system bandwidth to
communicate the same information in a digital format as compared to analog format.

2. System Synchronization:- Digital detection requires system synchronization whereas

the analog signals generally have no such requirement.

1. Channels for Digital Communications The modulation and coding used in a

digital communication system depend on the characteristics of the channel. The
two main characteristics of the channel are BANDWIDTH and POWER. In
addition the other characteristics are whether the channel is linear or nonlinear, and
how free the channel is free from the external interference.

Five channels are considered in the digital communication, namely: telephone channels,
coaxial cables, optical fibers, microwave radio, and satellite channels. Telephone channel:
It is designed to provide voice grade communication. Also good for data communication
over long distances. The channel has a band-pass characteristic occupying the frequency
range 300Hz to 3400hz, a high SNR of about 30db, and approximately linear response.
For the transmission of voice signals the channel provides flat amplitude response. But
for the transmission of data and image transmissions, since the phase delay variations are
important an equalizer is used to maintain the flat amplitude response and a linear phase
response over the required frequency band. Transmission rates upto16.8 kilobits per
second have been achieved over the telephone lines.
Coaxial Cable: The coaxial cable consists of a single wire conductor centered inside an
outer conductor, which is insulated from each other by a dielectric. The main advantages
of the coaxial cable are wide bandwidth and low external interference. But closely spaced
repeaters are required. With repeaters spaced at 1km intervals the data rates of 274
megabits per second have been achieved.

Optical Fibers: An optical fiber consists of a very fine inner core made of silica glass,
surrounded by a concentric layer called cladding that is also made of glass. The refractive
index of the glass in the core is slightly higher than refractive index of the glass in the
cladding. Hence if a ray of light is launched into an optical fiber at the right oblique
acceptance angle, it is continually refracted into the core by the cladding. That means the
difference between the refractive indices of the core and cladding helps guide the
propagation of the ray of light inside the core of the fiber from one end to the other.
Compared to coaxial cables, optical fibers are smaller in size and they offer higher
transmission bandwidths and longer repeater separations.

Microwave radio: A microwave radio, operating on the line-of-sight link, consists

basically of a transmitter and a receiver that are equipped with antennas. The antennas are
placed on towers at sufficient height to have the transmitter and receiver in line-of-sight
of each other. The operating frequencies range from 1 to 30 GHz.

Under normal atmospheric conditions, a microwave radio channel is very reliable and provides path for
high-speed digital transmission. But during meteorological variations, a severe degradation occurs in the
system performance. Satellite Channel: A Satellite channel consists of a satellite in geostationary orbit,
an uplink from ground station, and a down link to another ground station. Both link operate at
microwave frequencies, with uplink the uplink frequency higher than the down link frequency. In
general, Satellite can be viewed as repeater in the sky. It permits communication over long distances at
higher bandwidths and relatively low cost.

Bandwidth: Bandwidth is simply a measure of frequency range. The range of frequencies contained in
a composite signal is its bandwidth. The bandwidth is normally a difference between two numbers. For
example, if a composite signal contains frequencies between 1000 and 5000, its bandwidth is 5000 -
1000, or 4000. If a range of 2.40 GHz to 2.48 GHz is used by a device, then the bandwidth would be
0.08 GHz (or more commonly stated as 80MHz).It is easy to see that the bandwidth we define here is
closely related to the amount of data you can transmit within it - the more room in frequency space, the
more data you can fit in at a given moment. The term bandwidth is often used for something we should
rather call a data rate, as in “my Internet connection has 1 Mbps of bandwidth”, meaning it can transmit
data at 1 megabit per second.

2.Explain the geometric representation of signals.

3.Explain Gram Schmidt orthogonalisation procedure
4.Mathematical models of communication system.
UNIT II – Base band formatting techniques
2 marks
1. Define Dirac comb or ideal sampling function. What is its Fourier Transform?
Dirac comb is nothing but a periodic impulse train in which the impulses
are spaced by a time interval of Ts seconds. The equation for the function is given

2. Give the interpolation formula for the reconstruction of the original signal g(t)
 from the sequence of sample values {g(n/2W)}.

3. State sampling theorem.

If a finite –energy signal g(t) contains no frequencies higher than W
hertz ,it is completely determined by specifying its co=ordinates at a
sequence of points spaced 1/2W seconds apart.
If a finite energy signal g(t) contains no frequencies higher than W hertz, it
may be completely recovered from its co=ordinates at a sequence of points
spaced 1/2W seconds apart.

4. Define quadrature sampling.

Quadrature sampling is used for uniform sampling of band pass signals
5. What is aliasing?
The phenomenon of a high-frequency in the spectrum of the original
signal g(t) seemingly taking on the identity of a lower frequency in the spectrum
of the sampled signal g(t) is called aliasing or foldover.

6. Give the expression for aliasing error and the bound for aliasing error.

respectively and then suppressing the sum-frequency components by means of

appropriate low pass filter. Under the assumption that fc>W,we find that

gI(t)&gQ(t) are both low-pass signals limited to -W<f<W. Accordingly each

component may be sampled at the rate of 2W samples per second. This type of
sampling is called quadrature sampling.

7. What is meant by PCM?

Pulse code modulation (PCM) is a method of signal coding in which the
message signal is sampled, the amplitude of each sample is rounded off to the
nearest one of a finite set of discrete levels and encoded so that both time and
amplitude are represented in discrete form.. This allows the message to be
transmitted by means of a digital waveform.
8. Define quantizing process.
The conversion of analog sample of the signal into digital form is called
quantizing process.
9. What are the two fold effects of quantizing process.
1. The peak-to-peak range of input sample values subdivided into a finite set of
decision levels or decision thresholds
2. The output is assigned a discrete value selected from a finite set of
representation levels are reconstruction values that are aligned with the treads
of the staircase.

10. What is meant by idle channel noise?

Idle channel noise is the coding noise measured at the receiver output with
zero transmitter input.

11. What is meant by prediction error?

The difference between the actual sample of the process at the time of
interest and the predictor output is called a prediction error.

12. Define delta modulation.

Delta modulation is the one-bit version of differential pulse code
modulation.

13. Define adaptive delta modulation.

The performance of a delta modulator can be improved significantly by
making the step size of the modulator assume a time- varying form. In particular,
during a steep segment of the input signal the step size is increased. Conversely,
when the input signal is varying slowly, the step is reduced , In this way, the step
size is adapting to the level of the signal. The resulting method is called adaptive
delta modulation (ADM).
14. Name the types of uniform quantizer?
1. Mid tread type quantizer.
2. Mid riser type quantizer.

15. Define mid tread quantizer?

Origin of the signal lies in the middle of a tread of the staircase.
Output

3a
2a
a Overload level
-3a/2 -a/2
Input
-a
-2a

Peak to peak excursion where a=delta

16. Define mid-riser quantizer?

Origin of the signal lies in the middle of a riser of the staircase

O/p

3a/2
a/2 Over load level

a 2a 3a 4a i/p
 17. Define quantization error?
Quantization error is the difference between the output and input values of
quantizer.

18. What you mean by non-uniform quantization?

Step size is not uniform. Non-uniform quantizer is characterized by a step
size that increases as the separation from the origin of the transfer characteristics is
increased. Non-uniform quantization is otherwise called as robust quantization
.
19. Draw the quantization error for the mid tread and mid-rise type of quantizer?
For mid tread type:
Quantization
error a/2

Input

-a/2 a

For mid riser type:

Quantization error

a/2

Input

a
20. What is the disadvantage of uniform quantization over the non-
uniform quantization?

SNR decreases with decrease in input power level at the uniform

quantizer but non-uniform quantization maintains a constant SNR for wide
range of input power levels. This type of quantization is called as robust
quantization.

21. What do you mean by companding? Define compander.

The signal is compressed at the transmitter and expanded at the receiver. This is
called as companding. The combination of a compressor and expander is called
a compander.

22. Draw the block diagram of compander? Mention the types of

companding? Block diagram:

Input Compressor uniform quantizer expander o/p signal

Transmitter receiver
Types of companding:
1. µ law companding
2. A law companding

23. What is PAM?

PAM is the pulse amplitude modulation. In pulse amplitude
modulation, the amplitude of a carrier consisting of a periodic train of
rectangular pulses is varied in proportion to sample values of a message
signal.
24. What is the need for speech coding at low bit rates?

The use of PCM at the standard rate of 64 Kbps demands a high channel
bandwidth for its transmission ,so for certain applications, bandwidth is at
premium, in which case there is a definite need for speech coding at low bit
rates, while maintaining acceptable fidelity or quality of reproduction.

25. Define ADPCM.

It means adaptive differential pulse code modulation, a combination of
adaptive quantization and adaptive prediction. Adaptive quantization refers to a
quantizer that operates with a time varying step size. The autocorrelation
function and power spectral density of speech signals are time varying functions
of the respective variables. Predictors for such input should be time varying. So
adaptive predictors are used.

26. What is meant by forward and backward estimation?

AQF: Adaptive quantization with forward estimation. Unquantized
samples of the input signal are used to derive the forward estimates.
AQB: Adaptive quantization with backward estimation. Samples of
the quantizer output are used to derive the backward estimates.
APF: Adaptive prediction with forward estimation, in which
unquantized samples of the input signal are used to derive the forward
estimates of the predictor coefficients.
APB: Adaptive prediction with backward estimation, in which Samples
of the quantizer output and the prediction error are used to derive estimates of
the predictor coefficients.

27. What are the limitations of forward estimation with backward stimation?
Side information
 Buffering
Delay

28. How are the predictor coefficients determined?

For the adaptation of the predictor coefficients the least mean
square (LMS) algorithm is used.
29. Define adaptive subband coding?
It is a frequency domain coder, in which the speech signal is divided in to
number of subbands and each one is coded separately. It uses non masking
phenomenon in perception for a better speech quality. The noise shaping is done
by the adaptive bit assignment.

30. What are formant frequencies?

In the context of speech production the formant frequencies are the
resonant frequencies of the vocal tract tube. The formants depend on the shape
and dimensions of the vocal tract.

31. What is the bit rate in

ASBC? Nfs= (MN)
(fs/M) Nfs->bit rate
M->number of subbands of equal
bandwidths N->average number of bits
fs/M->sampling rate for each subband

32. Define Adaptive filter?

It is a nonlinear estimator that provides an estimate of some desired
response without requiring knowledge of correlation functions, where the
filter coefficients are data dependent. A popular filtering algorithm is the LMS
algorithm.
 33. Define data Signalling Rate.

Data signalling rate is defined as the rate measured in terms bits per
second(b/s) at which data are transmitted.
Data signaling rate Rb=I/Tb Where
Tb=bit duration.

16 marks.
1.State and prove Sampling theorem.
SAMPLING:

Sampling Theorem for strictly band - limited signals

1.a signal which is limited to W f W , can be completely
n
described by g ( ) .
2W
n
2.The signal can be completely recovered from g( )
2W
Nyquist rate 2W
Nyquist interval 1
2W
When the signal is not band - limited (under sampling)
aliasing occurs .To avoid aliasing, we may limit the
signal bandwidth or have higher sampling rate.

Let g (t ) denote the ideal sampled signal

g (t ) g ( nTs ) (t nTs ) (3.1)

n

where Ts : sampling period

fs 1 Ts : sampling rate
From Table A6.3 we have

g( t ) (t nTs )
n

1 m
G( f ) (f )
Ts m Ts

f sG ( f mf s )
m

g (t ) fs G( f mf s ) (3.2)
m

or we may apply Fourier Transform on (3.1) to obtain

G (f) g ( nTs ) exp( j 2 nf Ts ) (3.3)

n

or G ( f ) f sG ( f ) fs G( f mf s ) (3.5)
m
m 0

If G ( f ) 0 for f W and Ts 1
2W
n j nf
G (f) g( ) exp( ) (3.4)
n 2W W
 n
To reconstruct g (t ) from g ( ) , we may have
2W
g (t ) G ( f ) exp( j 2 ft )df
W 1 n j nf
g( ) exp( ) exp( j 2 f t)df
W 2W n 2W W
n 1 W n
g( ) exp j 2 f (t ) df (3.8)
n 2W 2W W 2W
n sin( 2 Wt n )
g( )
n 2W 2 Wt n
n
g() sin c( 2Wt n ) , - t (3.9)
n 2W
(3.9) is an interpolat ion formula of g (t )

Figure 3.3 (a) Spectrum of a signal. (b) Spectrum of an undersampled

version of the signal exhibiting the aliasing phenomenon.
 Figure 3.4 (a) Anti-alias filtered spectrum of an information-bearing signal. (b) Spectrum of
instantaneously sampled version of the signal, assuming the use of a sampling rate greater than the
Nyquist rate. (c) Magnitude response of reconstruction filter

2.Explain Quantisation in detail

Quantization Process:

Define partition cell

J k : mk m mk 1 , k 1,2,, L (3.21)
Where mk is the decision level or the decision threshold .
Amplitude quantizati on : The process of transforming the
sample amplitude m( nTs ) into a discrete amplitude
( nTs ) as shown in Fig 3.9
If m(t ) J k then the quantizer output is νk where νk , k 1,2,, L
are the representa tion or reconstruction levels , mk 1 mk is the step size.
The mapping g( m) (3.22)
is called the quantizer characteri stic, which is a staircase function.
Figure 3.10 Two types of quantization: (a) midtread and (b) midrise.

Quantization Noise:
 Figure 3.11 Illustration of the quantization process

Let the quantizati on error be denoted by the random

variable Q of sample value q
q m (3.23)
Q M V , ( E[ M ] 0) (3.24)
Assuming a uniform quantizer of the midrise type
2m m ax
the step - size is (3.25)
L
m m ax m m m ax , L : total number of levels
1
, q
f Q (q) 2 2 (3.26)
0, otherwise
2 1
Q E[Q 2 ] 2
q 2 f Q (q )dq 2
q 2 dq
2 2
2
(3.28)
12
When the quatized sample is expressed in binary form,
L 2R (3.29)
where R is the number of bits per sample
R log 2 L (3.30)
2m m ax
(3.31)
2R
2 1 2
Q mmax 2 2 R (3.32)
3
Let P denote the average power of m(t )
P
( SNR ) o 2
Q

3P 2 R
( 2
)2 (3.33)
mmax
(SNR) o increases exponentia lly with increasing R (bandwidth ).
3.Explain PCM in detail.
5.Explain DPCM in detail.
6.Explain Delta modulation in detail.
7.Explain ADM in detail.
8.Compare all the digital modulation techniques.
9.Explain Linear prediction in detail.
Linear Prediction (to reduce the sampling rate):

Consider a finite-duration impulse response (FIR)

discrete-time filter which consists of three blocks :

1. Set of p ( p: prediction order) unit-delay elements (z-1)

2. Set of multipliers with coefficients w1,w2,…wp

3. Set of adders ( )

The filter output (The linear predition of the input ) is

p
xˆ n wk x(n k ) (3.59)
k 1

The prediction error is

e n x n xˆ n (3.60)
Let the index of performance be
J E e 2 n (mean square error) (3.61)
Find w1 , w2 , , w p to minimize J
From (3.59) (3.60) and (3.61) we have
p
2
J Ex n 2 wk E x n x n k
k 1
p p
w j wk E x n jxn k (3.62)
j 1 k 1
 Assume X (t ) is stationary process with zero mean ( E[ x[n]] 0)
2
X E x2 n (E x n )2
E x2 n
The autocorrelation
RX ( kTs ) RX k E xnxn k
We may simplify J as
p p p
2
J X 2 wk RX k w j wk RX k j (3.63)
k 1 j 1 k 1
p
J
2 RX k 2 w j RX k j 0
wk j 1
p
w j RX k j RX k RX k , k 1,2 , ,p (3.64)
j 1

(3.64) are called Wiener - Hopf equations

as , if R X1 exists w0 R X1rX (3.66)

T
where w0 w1 , w2 ,, w p
rX [ RX [1], RX [2],..., RX [ p]]T
RX 0 RX 1  RX p 1
RX 1 RX 0  RX p 2
RX
  
RX p 1 RX p 2  RX 0

RX 0 , RX 1 ,, RX p
Substituti ng (3.64) into (3.63) yields
p p
2
J min X 2 wk RX k wk RX k
k 1 k 1
p
2
X wk RX k
k 1
2
X rXT w 0 2
X rXT R X1rX (3.67)
 rXT R X1rX 0, J min is always less than 2
X
 Linear adaptive prediction :

The predictor is adaptive in the follow sense

1. Compute wk , k 1,2,, p, starting any initial values
2. Do iteration using the method of steepest descent
Define the gradient v ector
J
gk , k 1,2, ,p (3.68)
wk
wk n denotes the value at iteration n . Then update wk n 1
1
wk n 1 g k , k 1,2, ,p (3.69)
wk n
2
1
where is a step - size parameter and is for convenience
2
of presentation.
P
J
gk 2 RX k 2 w j RX k j
wk j 1
p
2E x n x n k 2 w j E x n j x n k , k 1,2,, p (3.70)
j 1

To simplify t he computing we use x n x n k for E[x[n]x[n - k]]

(ignore the expectatio n)
p
gˆ k n 2 x n x n k 2 w j n x n j x n k , k 1,2,, p (3.71)
j 1

p
wˆ k n 1 wˆ k n xn k xn wˆ j n x n j
j 1

wˆ k n x n k e n , k 1,2,, p (3.72)
p
where e n x n wˆ j n x n j by (3.59) (3.60) (3.73)
j 1

The above equations are called lease - mean - square algorithm

Figure 3.27

Block diagram illustrating the linear adaptive prediction process