54284953 Linear Prediction for Speech Encoding

8/2/2019 54284953 Linear Prediction for Speech Encoding

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Linear Prediction using Lattice Filters

and its application in Speech

Compression

Abstract

The project aims at studying the theory of Linear Prediction of Stationary Signals and

using this knowledge in the application of compression of Speech. Linear Prediction can

be achieved using a number of methods. We have used the Lattice Filter method of Linear

Prediction and highlighted the advantages of this method. The calculation of p

prediction coefficient involves the inversion of a P X P matrix. This involves O(p3)

operations. To reduce the number of operations we have used the Levinson Durbin

algorithm which exploits a special property of the autocorrelation matrix to reduce the

number of operations to O(p2).

To implement this technique on speech signal, we have segmented and windowed the

speech samples in order to treat it as a stationary signal. We have analyzed the forward

prediction error and the recreated signal on several criterion such as the order of prediction, size of the window segment and number of bits used to encode the error

sequence. We have also created compressed sound files which can be heard to get an

idea of the result obtained.


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Contents

1. Introduction..................................................................................................................... 51.1 Human Speech Production........................................................................................ 5

1.2 Theory of Speech Coding ......................................................................................... 7

1.3 Historical Perspective of Linear Predictive Coding.................................................. 92. Linear Prediction........................................................................................................... 11

2.1 Innovations representation of a Random Process................................................... 122.2 Rational Power Spectra........................................................................................... 142.3 Relationships between the Filter Parameters and the Autocorrelation Sequence... 16

2.4 Theory of linear Prediction ..................................................................................... 18

2.4.1 The Autocorrelation Method............................................................................ 19

2.4.2 The Covariance Method................................................................................... 20

3. Lattice Filters ................................................................................................................ 243.1 Prediction Model Order Selection .......................................................................... 29

4. The Levinson-Durbin Algorithm................................................................................. 315. Progress Report. ............................................................................................................ 36

6. Observations and Results.............................................................................................. 44

6.1 Effect of order of prediction ................................................................................... 446.1.1 Spectrum of the Error Signal ........................................................................... 47

6.2 Effect of segment size............................................................................................. 54

6.3 Effect of quantisation and compression of error signal. ......................................... 567.1 Conclusions................................................................................................................. 64

7.2 Future Scope of the Project......................................................................................... 65

8. References..................................................................................................................... 66A. Appendix I: New MATLAB Functions. ...................................................................... 68B. Appendix II: MATLAB Codes .................................................................................... 83

C. Appendix III C- Codes ............................................................................................. 90

Code C.1 Levinson Durbin Algorithm.......................................................................... 90Code C.2 Levinson Durbin Header File ....................................................................... 93

Code C.3 MA Lattice Filter .......................................................................................... 94

Code x.3 MA Lattice Filter Header File ....................................................................... 95Code C.3 AR Lattice Filter ........................................................................................... 96

Code C.3 AR Lattice Filter Header File ....................................................................... 98

Code C.3 Segmentation and Hanning Window ............................................................ 99

Code C.3 Segmentation and Hanning Window Header File ...................................... 101


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List of Figures

Figure 1.1: Human Speech Production System 8

Figure 1.2: Model of Speech Production...9Figure 2.1: Time and Power Spectral Density Representation14

Figure 2.2: Filters for generating random process from white noise...16Figure 3.1: Forward Linear Prediction 27Figure 3.2: Prediction Error Filter... 27

Figure 3.3: Single stage Lattice Filter. 28

Figure 3.4: Two stage Lattice Filter 29

Figure 3.5: P stage Lattice Filter. 30

Figure 5.1: Original, Error and Recreated signal (without segmentation).. 40Figure 5.2: Original, Error and Recreated signal (with non-overlapping segmentation) 41

Figure 5.3: Original, Error and Recreated signal (with overlapping windowing)42Figure 5.4: Original, Error and Recreated signal with lattice filters 43

Figure 5.5: Frequency Spectrum of Original Signal 44

Figure 5.6: Frequency Spectrum of Recreated Signal. 45Figure 6.1: Graph of Prediction Gain Vs Order of Prediction..49

Figure 6.2: Frequency representation of error when order p=2.. 50

Figure 6.3: Frequency representation of error when order p=6.. 51Figure 6.4: Frequency representation of error when order p=8. .52

Figure 6.5: Frequency representation of error when order p=12.52

Figure 6.6: Frequency representation of error when order p=20 52Figure 6.7: Frequency representation of error when order p=40 53Figure 6.8: Frequency representation of original signal. 54

Figure 6.9: Frequency representation of original signal (Shifted).. 55

Figure 6.10: Frequency representation of recreated signal (Shifted) p=856Figure 6.11: Frequency representation of recreated signal (Shifted) p=12. 56

Figure 6.12: Window size Vs Predictive Gain 58

Figure 6.13: Original Signal encoded in 8 bits 59Figure 6.14: Recreated Signal when error is encoded in 8 bits 60

Figure 6.15: Recreated Signal when error is encoded in 7 bits.... 60

Figure 6.16: Recreated Signal when error is encoded in 6 bits.61

Figure 6.17: Recreated Signal when error is encoded in 5 bits 61Figure 6.18: Recreated Signal when error is encoded in 4 bits.62

Figure 6.19: Recreated Signal when error is encoded in 3 bits.62Figure 6.20: Recreated Signal Spectrum when error is encoded in 8 bits 63

Figure 6.21: Recreated Signal Spectrum when error is encoded in 7 bits.63

Figure 6.22: Recreated Signal Spectrum when error is encoded in 6 bits.64

Figure 6.23: Recreated Signal Spectrum when error is encoded in 5 bits 64Figure 6.24: Recreated Signal Spectrum when error is encoded in 4 bits.65

Figure 6.25: Recreated Signal Spectrum when error is encoded in 3 bits 65


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Chapter 1

INTRODUCTION


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1. Introduction

Linear prediction modelling is used in a diverse area of applications, such as data

forecasting, speech coding, video coding, speech recognition, model-based interpolation,

signal restoration and impulse/step event detection. In our project we would be studying

and implementing linear predictive coding (lpc) for speech compression.

1.1 Human Speech Production

Regardless of the language spoken, all people use relatively the same anatomy to produce

sound. The output produced by each humans anatomy is limited by the laws of physics.

The process of speech production in humans can be summarized as air being pushed from

the lungs, through the vocal tract, and out through the mouth to generate speech. In this

type of description the lungs can be thought of as the source of the sound and the vocal

tract can be thought of as a filter that produces the various types of sounds that make up

speech. The above is a simplification of how sound is really produced.

Figure 1.1: Human Speech Production System.


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Phonemes are defined as a limited set of individual sounds. There are two categories of

phonemes, voiced and unvoiced sounds. Voiced sounds are usually vowels and often

have high average energy levels and very distinct resonant or formant frequencies.

Voiced sounds are generated by air from the lungs being forced over the vocal cords. As

a result the vocal cords vibrate in a somewhat periodically pattern that produces a series

of air pulses called glottal pulses. The rate at which the vocal cords vibrate is what

determines the pitch of the sound produced. Unvoiced sounds are usually consonants and

generally have less energy and higher frequencies then voiced sounds. The production of

unvoiced sound involves air being forced through the vocal tract in a turbulent flow.

During this process the vocal cords do not vibrate, instead, they stay open until the soundis produced.

The amount of air that originates in the lungs also affects the production of sound in

humans. The air flowing from the lungs can be thought of as the source for the vocal tract

which act as a filter by taking in the source and producing speech. The higher the volume

of air that goes through the vocal tract, the louder the sound.

Figure 1.2: Model of Speech Production.

Pitch period

Vocal Tract

Parameters

Impulse Trainenerat r

Random Noise

enerat r

Voiced / Unvoiced switch

X(n)Time-Varying

Digital Filter

G


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Some of the fundamental properties of the speech signal that can be successfully

exploited for compression of speech include the quasi-stationary nature of the speech

signal. Quasi-stationary means that speech can be treated as a stationary signal for short

intervals of time. This allows us to use techniques which are generally used for stationary

signals for processing speech signals. The amplitude of speech signal varies slowly with

time, which is another characteristic that is commonly exploited for compression

purpose.

1.2 Theory of Speech Coding

The recent exponential growth of telecommunications today drives all aspects of

technology to higher degrees than ever before, creating the need to transfer maximal

amount of information consuming minimal relevant resources. Due to the impact on

parameters such as bandwidth requirements and conversation quality, the most important

component of any telephony system is that which generates the digital representation of

the speech.

Linear Predictive Coding (LPC) is defined as a digital method for encoding an analog

signal in which a particular value is predicted by a linear function of the past values of

the signal. It was first proposed as a method for encoding human speech by the United

States Department of Defense in federal standard 1015, published in 1984.

There exist many different types of speech compression that make use of a variety of

different techniques. However, most methods of speech compression exploit the fact that

speech production occurs through slow anatomical movements and that the speech

produced has a limited frequency range. The frequency of human speech production

ranges from around 300 Hz to 3400 Hz. Speech compression is often referred to as

speech coding which is defined as a method for reducing the amount of information

needed to represent a speech signal. There are many other characteristics about speech

production that can be exploited by speech coding algorithms. One fact that is often used

is that period of silence take up greater than 50% of conversations. An easy way to save


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bandwidth and reduce the amount of information needed to represent the speech signal is

to not transmit the silence. Another fact about speech production that can be taken

advantage of is that mechanically there is a high correlation between adjacent samples of

speech. Most forms of speech compression are achieved by modeling the process of

speech production as a linear digital filter. The digital filter and its slow changing

parameters are usually encoded to achieve compression from the speech signal.

Any signal processing system that aims to achieve utmost economy in the digital

representation of speech for storage or transmission must be based on the physical

constraints of our speech production apparatus and must exploit the limitations of humanhearing. It is wasteful to reserve costly bits for signals that the human mouth (and nose)

can never emit; it is equally wasteful to represent signal differences in the encoded bit

stream that the human ear can never distinguish.

Speech coding or compression is usually conducted with the use of voice coders or

vocoders. There are two types of voice coders: waveform-following coders and model-

base coders. Waveform following coders will exactly reproduce the original speech

signal if no quantization errors occur. Model-based coders will never exactly reproduce

the original speech signal, regardless of the presence of quantization errors, because they

use a parametric model of speech production which involves encoding and transmitting

the parameters not the signal. LPC vocoders are considered model-based coders which

mean that LPC coding is lossy even if no quantization errors occur.

The general algorithm for linear predictive coding involves an analysis or encoding part

and a synthesis or decoding part. In the encoding, LPC takes the speech signal in blocks

or frames of speech and determines the input signal and the coefficients of the filter that

will be capable of reproducing the current block of speech. This information is quantized

and transmitted. In the decoding, LPC rebuilds the filter based on the coefficients

received. The filter can be thought of as a tube which, when given an input signal,

attempts to output speech. Additional information about the original speech signal is used


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by the decoder to determine the input or excitation signal that is sent to the filter for

synthesis.

1.3 Historical Perspective of Linear Predictive Coding

The history of audio and music compression begin in the 1930s with research into pulse-

code modulation (PCM) and PCM coding. Compression of digital audio was started in

the 1960s by telephone companies who were concerned with the cost of transmission

bandwidth. Linear Predictive Codings origins begin in the 1970s with the development

of the first LPC algorithms. Adaptive Differential Pulse Code Modulation (ADPCM),

another method of speech coding, was also first conceived in the 1970s.

The history of speech coding makes no mention of LPC until the 1970s. However, the

history of speech synthesis shows that the beginnings of Linear Predictive Coding

occurred 40 years earlier in the late 1930s. The first vocoder was described by Homer

Dudley in 1939 at Bell Laboratories. Dudley developed his vocoder, called the Parallel

Bandpass Vocoder or channel vocoder, to do speech analysis and re-synthesis. LPC is a

descendent of this channel vocoder. The analysis/synthesis scheme used by Dudley is the

scheme of compression that is used in many types of speech compression such as LPC.

The idea of using LPC for speech compression came up in 1966 when Manfred R.

Schroeder and B.S. Atal turned their attention to the following: for television pictures, the

encoding of each picture element (pixel) as if it was completely unpredictable is of

course rather wasteful, because adjacent pixels are correlated. Similarly, for voiced

speech, each sample is known to be highly correlated with the corresponding sample that

occurred one pitch period earlier. In addition, each sample is correlated with the

immediately preceding samples because the resonances of the vocal tract. Therefore short

durations of speech show an appreciable correlation.


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Chapter 2

LINEAR PREDICTION


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2. Linear Prediction

The success with which a signal can be predicted from its past samples depends on the

autocorrelation function, or equivalently the bandwidth and the power spectrum, of the

signal. As illustrated infigure 2.1, in the time domain, a predictable signal has a smooth

and correlated fluctuation, and in the frequency domain, the energy of a predictable

signal is concentrated in narrow band/s of frequencies. In contrast, the energy of an

unpredictable signal, such as white noise, is spread over a wide band of frequencies.

Figure 2.1: Time and Power Spectral Density Representation.

For a signal to have a capacity to convey information it must have a degree of

randomness. Most signals, such as speech, music and video signals are, partially

predictable and partially random. These signals can be modeled as the output of a filter

excited by an uncorrelated input. The random input models the unpredictable part of the

signal, whereas the filter models the predictable structure of the signal. The aim of linear

prediction is to model the mechanism that introduces the correlation in a signal.


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2.1 Innovations representation of a Random Process

A wide sense stationary random process may be represented as the output of a causal and

causally invertible linear system excited by a white noise process. The condition that the

system is causally invertible also allows us to represent the wide sense stationary random

process by the output of the inverse system, which is a white process. This statement is

explained below.

Let us consider a wide-sense stationary process x(n) with the autocorrelation sequence

)(mxx and power spectral density )(fxx

, | f|


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where by definition 2w =exp[v(0)]

and 11

,)(exp)( rzzmvzHm

m

>

=

=

On evaluating the above equation on the unit circle, we have the equivalent

representation of the power spectral density as

22 )()( fHf wxx = 2.4

The filter with the system function H(z) is analytic in the region |z|>r1


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2.2 Rational Power Spectra

Considering the power spectral density of the stationary random process x(n) is a rational

function, expressed as

,)()(

)()()(

1

12

=zAzA

zBzBz wxx 21 rzr 2.7

where bk and ak are the filter coefficients that determine the location of the zeros and

poles of H(z), respectively. Thus H(z) is causal, stable, and minimum phase. Its

reciprocal 1/H(z) is also a causal, stable, and minimum phase linear system. Therefore the

random process x(n) uniquely represents the statistical properties of the innovation

process w(n), and vice versa.

For the linear system with the rational system function H(z) given by above equation, the

outputx(n) is related to the input w(n) by the following difference equation

==

=+

q

k

k

p

k

k knwbknxanx01

)()()( 2.8


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Distinguishing among three special cases:

Autoregressive (AR) Process: b0 = 1, bk= 0, k > 0

In this case the linear filter H(z) = 1/A(z) is an all-pole filter and the difference equation

for the input-output relationship is

)()()(1

nwknxanxp

k

k =+=

2.9

In turn, the noise-whitening filter for generating the innovations process is an all zero

filter.

Moving Average (MA) process : ak= 0, k >= 1

In this case the linear filter H(z) = B(z) is an all-zero filter and the difference equation for

the input-output relationship is

=

=

q

k

k knwbnx0

)()( 2.10

The noise-whitening filter for generating the innovations process is an all pole filter.

Autoregressive, Moving Average (ARMA) Process : In this case the linear filter

H(z) = B(z)/A(z) has both poles and zeros in the z-plane and the corresponding difference

equation is given by 2.8. The inverse system for generating the innovations process w(n)

fromx(n) is also a pole-zero system of the form 1/H(z) = A(z)/B(z).


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2.3 Relationships between the Filter Parameters and theAutocorrelation Sequence

When the power spectral density of the stationary random process is a rational function,

there is a basic relationship that exists between the autocorrelation sequence)(mxx and

the parameters akand bkof the linear filter H(z) that generates the process by filtering the

white noise sequence w(n). This relationship may be obtained by multiplying the

difference equation in 2.8 by x*(n-m) and taking the expected value of the both sides of

the resulting equation, to get

==

+=

q

k

wxk

p

k

xxkxx kmbkmam01

)()()(

2.11

where)(mwx is the cross-correlation sequence between w(n) andx(n).

The cross-correlation sequence)(mwx is related to the filter impulse response, as shown

below

[ ])()()( * mnwnxEmwx +=

+=

=

)()()(0

* mnwknwkhEk 2.12

)(2 mhw =

where in the last step, it was assumed that the sequence w(n) is white.

From 2.12 the following relationship is obtained:


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=

=

=

+

=

0),(

0,)()(

),(

)(

*

1 0

2

1

mm

qmbkhkma

qmkma

m

xx

p

k

mq

k

mkwxxk

p

k

xxk

xx

2.13

This represents a nonlinear relationship between)(mxx and the parameters akand bk.

The relationship in 2.13 applies, in general, to the ARMA process. For an AR process,

2.13simplifies to

=

=

=

0),(

0,)(

0),(

)(

*

1

2

1

mm

mkma

mkma

m

xx

p

k

wxxk

p

k

xxk

xx

2.14

Thus a linear relationship is obtained between)(mxx and the parameters ak and bk.

These equations are called the Yule-Walker equations and may be expressed in the matrix

form

=

0

....0

0

....

1

)0(........)2()1(

........................

)2(........)0()1(

)1(........)1()0(

2

2

*

** w

p

xxxxxx

xxxxxx

xxxxxx

a

a

a

pp

p

p

2.15

This correlation matrix is Toeplitz and hence it can be efficiently inverted by the use of

Levinson-Durbin algorithm as shown later.


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2.4 Theory of linear Prediction

Linear prediction involves predicting the future values of a stationary random process

from the observation of past values of the process. Consider, in particular, a one step

forward linear predictor, which forms the prediction of the value x(n) by a weighted

linear combination of the past values x(n-1),x(n-2) ... x(n-p).

Hence linearly predicted value of x(n) is

)(^

nx = - )()(1

knxkapk

k

p =

=

2.16

Where the ap(k) represent the weights in the linear combination. These weights are

called prediction coefficients of the one step forward linear predictor of order P. The

negative sign in the definition of x(n) is for mathematical convenience.

The difference between the value x(n) and the predicted value )(^

nx is called the forward

prediction error, denoted by fp(n) ,

)()()(^

nxnxnfp

= = )(nx + )()(1

knxkapk

k

p =

=

2.17

For information bearing signals, the prediction error fp(n) may be regarded as the

information, or the innovation, content of the sample.

To calculate the optimum prediction coefficients for our prediction filter we minimize the

mean square error i.e.

( )()(^

nxnx )2

is minimum.


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Two approaches can obtain the LPC coefficients akcharacterizing an all-pole H(z) model.

The least mean square method selects ak to minimize the mean energy in e(n) over a

frame of signal data, while the lattice filter approach permits instantaneous updating of

the coefficients.

The first of the two common least squares technique is the autocorrelation method,

which multiplies the speech signal by a window w(n) so that x (n)=w(n)x(n) has a finite

duration.

The autocorrelation sequence rss describes the redundancy in the signal x(n).

2.18

wherex(n), n = {-P, (-P) + 1, . . . ,N - 1} are the known samples and the

N is a normalizing factor.

2.4.1 The Autocorrelation Method

In this method, the speech segment is assumed zero outside the interval .10 Nm

Thus the Speech Sample can be expressed as

( )( ) ( )

+

=otherwise

Nmmwnmxmxn

,0

10,.

2.19

Another least square technique called covariance method windows the error signal,

instead of the actual speech signal.


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Autocovariance measures the redundancy in a signal

2.20

2.4.2 The Covariance Method

An alternative to using a weighting function or window for defining xn(m) is to fix the

interval over which the mean squared error is computed to the range

10 Nm

And use the unweighted speech directly. That is

( )meEN

nn

=

1

0

2

where( )kin , is defined as

( ) ( ) ( )

=

=

1

0,,

N

m

nnn kmximxki pk

pi

0

1

2.21

or by a change of variable

( ) ( ) ( ) += kimxmxki nnn , pk

pi

0

1

2.22

Using the extended speech interval to define the covariance values, ( )ki, , the matrix

form of the LPC analysis equation becomes,


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( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )

=

p,0

:

2,0

1,0

:

pp,.........p,2p,1

:.........::

p2,.........2,22,1

p1,.........1,21,1

2

1

pa

a

a

2..23

The resulting covariance matrix is symmetric (since( ) ( )ikki nn ,, = ) but not Toeplitz,

and can be solved efficiently be a set of techniques called the Cholesky decomposition.

The mean-square value of the forward linear prediction errorfp(n) based on the

autocorrelation method is

= = =

++==

p

k

p

k

p

l

xxppxxpxxp

f

p klkalakkanfE1 1 1

**2 )()()()]()(Re[2)0(]|)([| 2.24

Now, fp is a quadratic function of the predictor coefficients, and its minimization leads

to the set of linear equations

)(lxx = - plklkap

k

xxp ,......,2,1.....,.........)()(1

==

2.25

These are called the normal equations for the coefficients of the linear predictor. The

minimum mean-square prediction error is thus :

)()()0(]min[1

kkaE xx

p

k

pxx

f

p

f

p += =

2.26

Writing eq 2.26 in terms of vectors

xxxx rR =pa f 2.27


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or which the predictor coefficients can be obtained as

xxxx rR1

p )(a

= 2.28

A question may arise as to whether to use the autocorrelation method or the covariance

method in estimating the predictor parameters. The covariance method is quite general

and can be used with no restrictions. The only problem is that of stability of the resulting

filter. In the autocorrelation method on the other hand, the filter is guaranteed to be

stable, but problems of the parameter accuracy can arise because of the necessity of the

windowing (truncating) the rime signal. This is usually a problem if the signal is a portion

of an impulse response. For example, if the impulse response of an all-pole filter is

analyzed by covariance method, the filter parameters can be computed accurately from

only a finite number of samples of the signal. Using the autocorrelation method, one can

not obtain the exact parameters values unless the whole infinite impulse response is used

in the analysis. However, in practice, very good approximations can be obtained by

truncating the impulse response at a point where most of the decay of the response has

already occurred.


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Chapter 3

LATTICE FILTERS


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3. Lattice Filters

Linear prediction can be viewed as being equivalent to linear filtering where the predictor

is embedded in the linear filter, as shown in figure 3.1.

Figure 3.1: Forward Linear Prediction.

This is called a prediction-error filterwith the input sequence x(n) and the output

sequencefp(n). An equivalent realization for the prediction-error filter is shown in fig 3.2

Figure 3.2: Prediction Error Filter.

x(n) p(n)

_

+

)(^

nxx(n-1)z

-1 Forward

Linear

Predictor

fp(n)

x(n)

ap(p)ap(p-1)ap(3)ap(2)ap(1)1

z-1 z-1 z-1 z-1


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This realization is a direct-form FIR filter with the system function given as

=

=

p

k

k

pp zkazA0

)()( 3.1

where, by definition, ap(0) = 1.

Prediction Error filters can be realised in other way also, which take the form of a lattice

structure. To find a relationship between the Lattice filter coefficients and the FIR filter

structure, let us begin with a predictor of order p=1. The output of such a filter is

)1()1()()( 11 += nxanxnf 3.2

This output can be obtained from the single stage lattice filter, as shown in figure 3.3

below, by exciting both the inputs byx(n) and selecting the output from the top branch.

Figure 3.3: Single stage Lattice Filter.

Thus the output is exactly that given by above equation if we select K1 = a1(1). The

parameterK1 in the lattice filter is called a reflection coefficient.

The negated reflection coefficient, - km , is also called the partial correlation (PARCOR)

coefficient

f0(n)

g0(n-1)g0(n) g1(n)

f1(n)

K

K

z-1

n


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Next, considering a predictor of order p = 2. For this case, the output of the direct-form

FIR filter is :

)2()2()1()1()()( 222 ++= nxanxanxnf 3.3

By cascading two lattice stages as shown in the figure 3.4, it is possible to obtain the

same output as above, as shown.

Figure 3.4: Two stage Lattice Filter.

The two outputs form the first stage are :

)1()()(

)1()()(

*

11

11

+=

+=

nxnxKng

nxKnxnf3.4

Similarly, the two outputs from the second stage are :

)1()()(

)1()()(

11

*

22

1212

+=

+=

ngnfKng

ngKnfnf3.5

Substituting the value off1(n) andg1(n-1) in above equations yield:

)2()1()()()( 22*

112 +++= nxknxkkknxnf 3.6

g2(n)

f2(n)

K2*

K

f0(n)

g0(n-1)

g0(n) g1(n)

f1(n)

K*

K

-1

x n

-1


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On equating coefficients we get,

a2(2)=K2 and a2(1)= )( 2*

11 kkk + 3.7

or equivalently,

K2 = a2(2), K1 = a1(1) 3.8

By continuing this process, the equivalence between an mth order direct-form FIR filter

and an mth stage lattice filter can be demonstrated. The lattice is described by the

following set oforder-recursive equations:

)1()()(

)1()()(

)()()(

11

*

11

00

+=

+=

==

ngnfKng

ngKnfnf

nxngnf

mmmm

mmmm

pm

pm

,......,2,1

,......,2,1

=

=

3.9

A p-stage lattice filter for pth

order predictor can be shown as follows :

Figure 3.5: P stage Lattice Filter.

As a consequence of the equivalence between the direct form prediction error filter and

the FIR lattice filter, the output of the p-stage lattice filter is expressed as :

=

==

p

k

ppp aknxkanf0

1)0(.....................),........()()( 3.10

gp(n)

fp(n)

g2(n)

f2(n)

g1(n)

f1(n)

g0(n)

f0(n)

First

Stage

Second

Stage

pth

Stage


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The lattice forms characterization requires only p reflection coefficients Ki for a p step

linear predictor in comparison with the p(p+1)/2 filter coefficients required by the FIR

filter implementation. The reason that the lattice provides a more compact representation

is because appending stages to the lattice does not alter the parameters of the previous

stages. On the other hand, appending the pth

stage to a FIR based predictor results in

system function Ap(z) that has coefficients totally different from the coefficients of the

lower-order FIR filter with system functionAp-1(z).

Although the direct-form implementation of the linear predictor is the most convenient

method, for many applications, such as transmission of the predictor coefficients inspeech coding, it is advantageous to use the lattice form of predictor. This is because the

lattice form can be conveniently checked for stability. That is, for a stable model, the

magnitude of reflection coefficient is bounded by unity, and therefore it is relatively easy

to check a lattice structure for stability.

The quantization of the filter coefficients for transmission can create a major problem

since errors in the filter coefficients can lead to instability in the vocal tract filter and

create an inaccurate output signal. This potential problem is averted by quantizing and

transmitting the reflection coefficients that are generated by the Levinson-Durbin

algorithm. These coefficients can be used to rebuild the set of filter coefficients {a i} and

can guarantee a stable filter if their magnitude is strictly less than one.

A major attraction of a lattice structure is its modular form and the relative ease with

which the model order can be extended. Furthermore a perturbation of the parameter of

any section of the lattice structure has a limited and more localized effect.


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3.1 Predictor Model Order Selection

One procedure for the determination of the correct model order is to increment the model

order, and monitor the differential change in the error power, until the change levels off.

The incremental change in error power with the increasing model order from i-1 to I is

defined as

)()1()( iii EEE = 3.11

The orderp beyond which the decrease in the error power becomes less than a thresholdis taken as the model order.

When the model order is less than the correct order, the signal is under-modelled. In this

case, the prediction error is not well decorrelated and will be more than the optima;

minimum. A further consequence of the under-modelling is a decrease in the spectral

resolution of the model: adjacent spectral peaks of the signal could be merged and appear

as a single spectral peak when the model order is too small. When the model order is

larger than the correct order, the signal is over-modelled. An over-modelled problem can

result in an ill-conditioned matrix equation, unreliable numerical solutions and the

appearance of spurious spectral peaks in the model.


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Chapter 4

Levinson Durbin Algorithm


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4. The Levinson-Durbin Algorithm

The Levinson Durbin algorithm is a computationally efficient algorithm for solving the

normal equations. It is named so in recognition of its first use by Levinson(1947) and

then its independent reformulation at a later date by Durbin(1960).

,0)()(0

==

klka xx

p

k

p l=1, 2, , p, ap(0)=1 4.1

for the prediction coefficients. This algorithm exploits the special symmetry in the

autocorrelation matrix

=

)0(........)2()1(

........................

)2(........)0()1(

)1(........)1()0(*

**

xxxxxx

xxxxxx

xxxxxx

p

pp

p

p

4.2

Since )(),( jiji pp = , so the autocorrelation matrix is a Toeplitz matrix. Also, since

),(),( * jiji pp = , the matrix is also Hermitian.

The key to the Levinson Durbin method of solution that exploits the Toeplitz property of

the matrix is to proceed recursively, beginning with a predictor of the order m=1 (one

coefficient) and to increase the order recursively, using the lower order solutions to

obtain the solution to the next higher order. Thus the solution to the first order predictor

obtained by solving the equation is

)0(/)1()1(1 xxxxa = 4.3

and the resulting minimum mean square error (MMSE) is


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]|)1(|1)[0( 211 aE xxf

= 4.4

The next step is to solve for the coefficients a2(1) and a2(2) of the second order predictor

and express the solution in terns of a1(1). The resulting equations are

)2()0()2()1()1(

)1()1()2()0()1(

22

*

22

xxxxxx

xxxxxx

aa

aa

=+

=+4.5

By using the solution in 4.4 to eliminate )1(xx , the following equations are obtained

])1(1)[0(

)2()1()1()2(

2

1

1

2

a

aa

xx

xxxx

+=

4.6

f

xxxx

E

a

1

1 )2()1()1( +=

)1()2()1()1( *1212 aaaa += 4.7

In this manner to represent in terms of a recursive equation we express the coefficients of

the mth order predictor in terms of the coefficients of the (m-1)st order predictor, We can

write the coefficient vector am as the sum of two vectors, namely

+

=

=

m

mm

m

m

m

m

m

K

da

ma

a

a

a

...

0

...

)(

....)3(

)2(

)1( 11

a 4.8

where am-1 is the predictor coefficient vector of the (m-1)st order predictor and the vector

dm-1 and the scalar Km are the scalar Km are to be determined. For this the m X m

autocorrelation matrix xx as


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=

)0(1

*

11

xxbtm

mm

m

4.9

where [ ] ( )tbmxxxxxxbt

m mm 11 )1(.......)2()1( == .The superscript b on 1m

denotes the vector [ ])1(.......)2()1(1 = mxxxxxxbt

m with elements taken in

reverse order.

The solution to the equation mmma = may be expressed as

=

+

)(0)0(

111

1

*

11

mK

da

xx

m

m

mm

xx

bt

m

mm

4.10

This is the key step in the Levinson Durbin algorithm. From 4.10, two equations are

obtained as follows

1

*

11111 =++ mb

mmmmmm Kda 4.11

)()0(11111 mKda xxxxmmbt

mm

bt

m =++ 4.12

Since 111 = mmm a , 4.11 yields the solution

*

1

1

11

b

mmmm Kd

=

==

)1(

...

)2(

)1(

*

1

*1

*

1

*

11

m

m

m

m

b

mmm

a

ma

ma

KaKd 4.13

The scalar equation 4.12 can be used to solve forKm


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( )

( ) * 11

11

0

+

=

m

bt

mxx

m

bt

mxx

ma

amK

4.14

Thus substituting the solutions in 4.13 and 4.14 into 4.8, we get the recursive equations

for the predictor coefficients and the reflection coefficients for the lattice filters. The

recursive equations are given as

( )( )

( )

( )

( ) ( )( ) ( ) ( ) )(

)(

0

*

11

*

11

11

*

11

11

kmamakakakmaKkaka

E

am

a

amKma

mmmm

mmmm

f

m

m

bt

mxx

m

bt

mxx

m

bt

mxx

mm

+=

+=

+=

+==

4.15

From the equations we note that the predictor coefficients form a recursive set of

equations. Km is the reflection of the mth

stage. Also Km=am(m), the mth coefficient of

the mth

stage.

The important virtue of the Levinson-Durbin algorithm is its computational efficiency, in

that its use results in a big saving in the number of operations. The Levinson-Durbin

recursion algorithm requires O(m) multiplications and additions(operations) to go from

stage m to stage m+1. Therefore, for p stages it will take on the order of 1+2+3++p =

p(p+1)/2, or O(p2) operations to solve for the prediction filter coefficients or the

reflection coefficients, compared with O(p3) operations if the Toeplitz property is not

exploited.


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Chapter 5

Progress Report


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5. Progress Report.

As required by the normal equations, we have to calculate the autocorrelation matrix and

solve the equation ap=(Rxx)-1 rxx. The first program made by us calculated the

autocorrelation matrix and compared the value of the prediction coefficients obtained by

using the matrix inversion function of MATLAB6p1 with the linear prediction function

of MATLAB6p1. The results are shown below.

Using inv() function Using lpc() function

1.0000 1.0000-1.6538 -1.6538

1.3089 1.3089

-1.0449 -1.0449

0.4729 0.4729

-0.0588 -0.0588

-0.1057 -0.1057

0.6557 0.6557

-0.8568 -0.8568

0.5507 0.5507

-0.2533 -0.2533

-0.0262 -0.0262

0.1345 0.1345

Table 5.1

Since both functions returned identical values, our autocorrelation matrix was correctly

calculated. Next we implemented the Levinson-Durbin algorithm using the recursive

relations as explained previously. The function made by us returned both, the prediction

coefficients and the reflection coefficients. Hence this function can be used to implement

the linear predictor using either FIR filter or lattice filter.

To check the accuracy of our function, we made a program that calculated the prediction

coefficients for a speech input signal and implemented FIR predictive filter to generate


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the error sequence. The same coefficients were then used to implement the inverse filter

that generated back the speech signal from the error signal. The figure below shows the

original, error and recreated signal. It is clear that the recreated signal is same as the

original input signal.

Figure 5.1: Original, Error and Recreated signal (without segmentation)

The above sample is for the utterance of the word N-S-I-T. The speech sample is encoded

in 8bits and sampled at a frequency of 11025Hz corresponding to telephone quality. In all

the examples in this section, 8th order Linear Prediction is used. The error signal is clearly

smaller than the original signal and will require fewer bits to encode. This program does

not use advanced techniques such as segmentation, windowing or end detection. But

since speech is only quasi stationary and cannot be assumed stationary for such large

number of samples we will have to perform segmentation, windowing of this speech


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sample. This will further reduce the magnitude of the error signal helping in improved

compression of the speech.

Next we divided the speech into non-overlapping segments and again calculated the error

and the recreated signal. The result is shown below. In this program we have used the

filter programs created by us. We have used a constant window size of 15ms which

corresponds to 166 samples.

Figure 5.2: Original, Error and Recreated signal (with non-overlappingsegmentation)

The figure shows that using non-overlapping segments cause spikes due to discontinuities

where errors are large. This is because we are trying to predict speech from 0 at the

edges. To overcome this limitation, overlapping segments of the speech are taken and

windowing is done. The most popular windows are the Hamming and Hanning windows.


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Below is shown the output when we take overlapping segments overlapping by N/2,

where N is the window size. Hanning window is used in this example.

Figure 5.3: Original, Error and Recreated signal (with overlapping segmentationand windowing)

It can be clearly seen from the above figure that the error signal has become smooth as

compared to the non-overlapping segmentation case. Hence this error signal can be

encoded successfully.


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As the final program in this preliminary case, we will solve the same case using lattice

filters instead of the FIR and IIR filters. Lattice filters have a number of advantages over

FIR/IIR implementation in the case of Linear Prediction. These have already been

explained earlier.

Figure 5.4: Original, Error and Recreated signal with lattice filters.

Hence we can see that Lattice Filters give the same result as the other filters.


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The spectrum of the input signal and that of the output signal is shown below. There is no

difference in the spectrum.

Figure 5.5: Frequency Spectrum of Original Signal.


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Figure 5.6: Frequency Spectrum of Recreated Signal.


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Chapter 6

Observation and Results


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6. Observations and Results

All software simulations have been performed on MATLAB6p1, which is a registered

trademark of The Mathworks Incorporation.

We have shown earlier how we have reached the stage where error is minimum by using

segmentation of speech and windowing. In this section we will show the effect of

changing various parameters such as order of prediction, size of window, on the error

signal and the recreated signal. To take into effect of quantization and compression, we

will encode the error signal in less number of bits before using it to recreate back the

speech signal. This will ensure that the observations are accurate.

The speech sample used for the entire observation will be the word N-S-I-T encoded in 8

bits sampled at a frequency of 11025 Hz, which corresponds to telephone quality. The

order of prediction will be 8 and the size of window 15ms corresponding to 166 samples

of speech. Three different encoding schemes will be used. The error signal will beencoded in 8 bits to 3 bits. During each of the sub-segment, one parameter will be varied

while all the remaining parameters will hold the value as specified above.

6.1 Effect of order of prediction

The order of prediction governs how many previous samples as used to predict the next

sample. As the order of the predictor is increased, up to the order of the process which

generated the signal, the power spectrum of the error signal will become more and more

flat. But it is not possible to increase the order arbitrary since the autocorrelation matrix is

a p X p matrix, where p is the order, and solving this matrix requires lot of computation

time. Through large number of experimentation and observation, it has been seen that an

order of 8-12 is suitable for most speech samples.


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In this section we will vary the order and see the effect on the error signal. We will

measure the predictive gain i.e. the 10 log (variance of the original signal / variance of the

error signal). A table of the observations is shown below.

Number of Bits

Order

8 7 6 5 4 3

2 3.8176 3.7277 3.4369 3.0975 2.0599 0.29527

46.7546 6.6762 6.3967 5.9849 4.8896 3.8079

6 9.7913 9.6751 9.3888 5.9849 7.9312 7.0689

8 11.113 11.037 10.683 10.246 9.1502 8.0151

10 13.269 13.157 12.78 12.402 11.391 12.291

12 15.264 15.145 14.808 14.236 13.505 14.385

20 15.493 15.378 15.019 14.556 13.964 14.67

40 15.225 15.099 14.768 14.368 13.502 14.271

60 15.539 15.421 15.099 14.607 13.835 14.6

Table 6.1: Predictive Gain Vs Order of Prediction.


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Plotting a graph for the above table.

Figure 6.1: Graph of Prediction Gain Vs Order of Prediction.

Index

8 Bits: Purple

7 Bits: Green6 Bits: Magenta

5 Bits: Black

4 Bits: Blue3 Bits: Red


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6.1.1 Spectrum of the Error Signal

As the order of prediction increases, the spectrum of the error signal is flattened since the

prediction filter removed all the correlation from the input signal and gives the output as

a nearly white noise sequence.

Below are shown a few plots of the Frequency domain representation of the error signal

as the order of the predictor is increased. Clearly the spectrum is gradually flattened as

the order is increased. After the order becomes greater than the order of the system that

generated the original signal, there would not be any more flattening of the error signal.

This is because we cannot predict with more accuracy than the actual system that

generated the signal.

Figure 6.2: Frequency representation of error when order p=2..


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Figure 6.3: Frequency representation of error when order p=6.



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From the above figures we can see there is not much flattening of the error after order 20.

Generally for Speech signal an order of 12 is sufficient and gives good results.


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Shown below are the frequency spectrum of the original signal and the recreated signal.

Figure 6.8: Frequency representation of original signal.

This figure shows the frequency content of the original signal. Let us represent the

frequency spectrum in a more convenient form by using the function fftshift().


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Figure 6.9: Frequency representation of original signal (Shifted).

Below we show the frequency response of the recreated signal.


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.Figure 6.10: Frequency representation of recreated signal (Shifted) p=8.

Figure 6.11: Frequency representation of recreated signal (Shifted) p=12.


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6.2 Effect of segment size

Window size determines the number of segments that the speech is divided into. Since

speech is quasi stationary, it can be assumed stationary for only a small duration. The

smaller the segment size, the more number of segments will be created and the program

will require more computation time. But as the segments size is increased, the predictive

gain is reduced and hence compression is not very efficient. Hence we have to find a

balance between the size of the segment and the computation time required.

Number of Bits

Window

Size (ms)

8 7 6 5 4 3

5 11.956 11.832 11.499 11.024 9.9649 9.7677

7 11.164 11.076 10.705 10.294 9.4267 8.2087

10 11.471 11.4 10.996 10.569 9.4888 8.6293

13 11.032 10.935 10.618 10.07 9.2504 8.0782

15 11.113 11.037 10.683 10.246 9.1502 8.0151

17 11.035 10.944 10.576 10.153 9.1392 8.0709

20 10.945 10.829 10.492 9.9965 9.0453 7.9843

23 10.971 10.878 10.51 10.048 9.0963 8.0501

25 10.984 10.891 10.514 10.085 9.1083 7.9189

30 11.03 10.943 10.574 10.097 9.0816 8.1633

50 11.08 10.985 10.636 10.136 9.2119 8.3669

Table 6.2 Predictive Gain vs Window Segment Size


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Plotting the above table in a graph form.

Figure 6.12: Window size Vs Predictive Gain.

Index

8 bits: Red circles7 bits: Blue circles

6 bits: Magenta Triangles

5 bits: Green Stars4 bits: Blue Squares

3 bits: Black V


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6.3 Effect of quantization and compression of error signal.

Once the error signal has been quantized into less number of bits, some information has

been lost and cannot be achieved back. We have to concentrate on how best we can

reproduce the original signal from this quantized forward prediction error. Below are

shown the recreated signals for the same word N-S-I-T we have used earlier. The original

signal is encoded in 8 bits. We will encode the forward prediction error in 7, 6, 5, 4, 3 bits

and see the output waveform.

Figure 6.13: Original Signal encoded in 8 bits.


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Figure 6.14: Recreated Signal when error is encoded in 8 bits.



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The loss of quality of speech as the error signal is encoded in less and less bits is obvious.


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Figure 6.20: Recreated Signal Spectrum when error is encoded in 8 bits.



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Chapter 7

Conclusions and Future Scope of the Project


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7.1 Conclusions

From our experiments we have verified the accuracy and the efficiency of the Levinson

Durbin algorithm and its application to Linear Prediction. We have also clearly shown

that it is possible to compress speech signals in the form of forward prediction error

together with the filter parameters. If the error sample is stored in sufficient number of

bits then the recreated signal is of good quality.

As we have shown above we have managed to encode 8 bit speech sample into a 4 bit

error value while maintaining intelligibility of the speech. Depending on the application

and the quality of signal required we can choose the number of bits of precision to store

the error signal.

Our results show that use of windowing and segmentation is very useful in order to have

an efficient compression of the error signal.

The simulations results show that the prediction order in the range of 8-12 is sufficient

for linear prediction of speech, and using order above this does not give much

improvements.

Figures obtained by our simulations on MATLAB6.1 show that a window size of 5-10

msec gives very good results for speech samples at 11025 Hz. For good results, the

length of the window should contain ideally 60 to 100 speech samples.


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7.2 Future Scope of the Project

In our project we successfully simulated compression of speech in MATLAB. We also

made all the relevant functions on our own so that all the work can be easily implemented

in Hardware.

Future students working on this project will find the hardware implementation of our

project very simple as we have provided a ready made set of important functions along

with the source code. These functions can be easily coded in C language from which it

can be converted into the assembly language of any hardware.

Future work on the project can also be done on improved DSP techniques to further

reduce error and make compression more efficient.

The Linear Predictor Coefficients can be calculated faster using Schur Algorithm on

parallel architecture, which is much more computationally efficient. Hence real-time

speech processing can be performed.


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8. References

[1] J.Makhoul, Linear Prediction: A tutorial review; Proc of IEEE, vol 63, pp 561-589,

April 1975.

[2] John D Markel and A H Gray, A linear prediction vocoder simulation based upon the

autocorrelation method, IEEE Tran. on Acoustics, speech and signal processing; Vol.

ASSP: 22, April 1974.

[3] L.R Rabiner, B S Atal amd M R Sambur, LPC prediction arror: analysis of its

variation with position of analysis frame, IEEE Tran. on Acoustics, speech and signalprocessing; Vol. ASSP: 25, Oct 1977.

[4] Advanced Digital Signal Processing, Proakis J.G, Rader M.C, Ling F, Nikias C.L,

Macmillan Publishing Company, New York, 1992. ISBN- 0-02-396841-9.

[5] J.Makhoul, Stable and efficient Lattice Methods for Linear Prediction., IEEE Tran.

On Acoustic Speech and Signal Processing, Vol. ASSP: 25, Oct 1977

[6] Douglas OShaughnessy, Linear Predictive Coding, IEEE Potentials, Feb-1988

[7] John E. Roberts and R. H. Wiggins, PIECEWISE LINEAR PREDICTIVE CODING

(PLPC), The MITRE Corporation, Bedford, Massachusetts 01730, May-1980

[8] M. A. Atashroo, Autocorrelation Prediction, Advanced Research Project Agency.

[9] Manfred R. Schroeder, Linear Predictive Coding of Speech: Review and Current

Directions, IEEE Communications Magazine, Aug-1985, Vol-23, No. 8, pp 54-61

[10] Advanced Digital Signal Processing and Noise Reduction, Vaseghi Saeed V, John

Wiley and Sons, 1996. ISBN-0-471-62692-9.

[11] L.R Rabiner and B.H Juang, Fundamentals of Speech Processing, Prentice Hall,

1993.

[12] Simon Haykin, Adaptive Filter Theory, 3rd

edition, Prentice Hall International, New-

Jersey, ISBN 0-13-397985-7

54284953 Linear Prediction for Speech Encoding

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