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COLOURED NOISE REMOVAL AND EQUALISING THE CHANNEL EFFECT FROM A NOISY AUDIO SIGNAL

G. Anudeep Reddy (EC08484)G. Madhuri (EC08485)

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INTRODUCTION

We want to transmit a song signal.

Song SignalTransmitt

er

ReceiverLoud

Speaker

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REMOVAL OF NOISE

The signal may be corrupted by Noise.

Song Signal

Transmitter

Receiver

Noise

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REMOVAL OF NOISE

The signal may be corrupted by Noise.

Song Signal

Transmitter

ReceiverLoud

Speaker

Noise

Filter

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WHITE NOISE

White noise is a signal (or process), having equal power in any band of a given bandwidth (power spectral density).

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COLORED NOISE

Based on Spectral density (power distribution in the frequency spectrum) we can distinguish different types of noise.

This classification by spectral density is given "color" terminology, with different types named after different colors.

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COLORED NOISE

The color names for these different types of sounds are derived from a loose analogy between the spectrum of frequencies of sound wave present in the sound and the equivalent spectrum of light wave frequencies.

That is, if the sound wave pattern of "blue noise" were translated into light waves, the resulting light would be blue, and so on.

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PINK NOISE

Similar to White Noise except the power density decreases 3 dB per octave as the frequency increases. In technical terms the density is inversely proportional to the frequency.

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BLUE NOISE

Similar to White Noise except the power density increases 3 dB per octave as the frequency increases. In technical terms the density is proportional to the frequency.

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GENERATION OF COLORED NOISE

Colored noise can be generated by passing the white noise through a shaping filter.

The response of the colored noise can be varied by adjusting the parameters of the shaping filter.

White Noise

Shaping Filter

Colored Noise

Noise_filter.m

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REMOVAL OF CHANNEL EFFECT

The signal may be corrupted by channel transfer function also.

Song Signal

Transmitter

Receiver

Noise

Channel

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REMOVAL OF CHANNEL EFFECT

The signal may be corrupted by channel transfer function also.

Song Signal

Transmitter

Receiver

Loud Speake

r

Noise

Filter

Channel

Equalizer

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EQUALIZER

The equalizer should have transfer function which is inverse of channel.

Where H(z) is the transfer function of equalizer and C(z) is the transfer function of channel.

But in most of the cases we do not know the transfer function of the channel, so we will adapt the equalizer transfer function using Learning algorithm.

)(

1)(

zCzH

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COMPLETE BLOCK DIAGRAM

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CHANNEL AND CHANNEL EQUALIZER

o A finite impulse response (FIR) filter is a type of a signal processing filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.

o This is in contrast to infinite impulse response (IIR) filters, which have internal feedback and may continue to respond indefinitely (usually decaying).

o The impulse response of an Nth-order discrete-time FIR filter, lasts for N+1 samples, and then dies to zero.

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o FIR filters can be discrete time or continuous-time, and digital or analog.

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For a discrete-time FIR filter, the output is a weighted sum of the current and a finite number of previous values of the input. The operation is described by the following equation, which defines the output sequence y[n] in terms of its input sequence x[n]:

o The channel can be considered as a discrete time digital FIR filter

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From the block diagram, it is evident that the optimal equalizer should have transfer function which is inverse of channel. Hence channel equalization is also known as inverse filtering.

Transfer function of Channel , C(z)= b0+ b1z-1+ b2z-2 + bnz-n

Transfer function of the Equalizer, H(z)= 1/C(z)

o Similarly the equalizer can be considered as an FIR filter(discrete time, digital FIR filter)

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AN ADAPTIVE LINEAR EQUALIZER

Z-1 Z-1 Z-1

∑

xk

xk-1 xk-2 xk-L+1

w0k w1k w(L-1)k

yk

w2k

There is an input signal vector,

a corresponding set of adjustable weights,

a summing unit, and a single out put signal.

11,0 ... Lxxx

110 ..., Lwww

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ADAPTIVE LINEAR EQUALIZER

o A procedure for adjusting or adopting the weights is called weight adjustment or adaptation procedure.

o The combiner is called linear because for fix setting of weights its output is a linear combination of the input components.

o The output of the combiner can be represented as

lk

L

llkk xwy

0

where denotes weight at instant.lkw thl thk

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If the weight and input vectors are expressed as

TKLkkk xxxX ][ )1(10

TKLkkk wwwW ][ )1(10

then the output is given by

kTkk wxy

The weights of the combiner are to be updated using various learning algorithms

Ravi Kumar Jatoth Department of ECE NITW

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LEARNING ALGORITHMS

LMS Algorithm RLS Algorithm Kalman Filter Neural Algorithm Fuzzy Logic System Optimization Algorithms

All the algorithms update the weights of the equalizer using different cost functions.

Ravi Kumar Jatoth Department of ECE NITW

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LEAST MEAN SQUARE ALGORITHM

❏ LMS: adaptive filtering algorithm having

two basic processes

✔ Filtering process, producing

1) output signal

2) estimation error

✔ Adaptive process, i.e., automatic

adjustment of filter tap weights

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LEAST MEAN SQUARE ALGORITHM

o LMS algorithm is one of the conventional techniques applied to channel equalization. The cost function is Mean Square Error (MSE). It updates the weights of the adaptive FIR filter based on the error obtained. The instantaneous error at any time-step 'k' can be represented as

e(k) = d(k) – y(k)

where d(k) delayed input reference is signal at time-step ‘k’, and 'y(k)’ is estimated output from equalizer.

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oThe equalizer filter's impulse response vector is adapted using the following equation,

w(k+1) = w(k) + 2µ.e(k).x(k) where µ is called ‘Convergence factor’ or ‘Learning

rate parameter’, (0 ≤ µ ≤ 1). x(k) Is input from transmitter at time-step 'k'.

o This procedure is repeated till the Mean Square Error (MSE) of the network approaches a minimum value.

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351M Digital Signal Processing

STABILITY OF LMS

More practical test for stability is

Larger values for step size Increases adaptation rate (faster adaptation) Increases residual mean-squared error

power signal input2

0

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Fig. 2 Adaptive filter using LMS algorithm

Z-1 Z-1 Z-1

∑

LMSAlgorithm

xk

xk-1 xk-2 xk-L+1

w0k w1k w(L-1)k

∑

ekyk

dk

-

+

w2k

T1 1k k k k Lx x x X the L-by-1 tap input vector.

T

0 1 1k k k L kw w w W the L-by-1 tap weight vector

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LMS BLOCK DIAGRAM

Z-1

Z-1

∑

Z-1

Z-1

∑

LMS

+

-

y (k)

e (k)

a0

a1

a2

a3

a4

d(k)

x(k)

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NUMERICAL EXAMPLE- CHANNEL EQUALIZATION

❏ Transmitted signal: random sequence of

±1’s.

❏ The transmitted signal is corrupted by a

channel.

❏ Channel impulse response:

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❏ The amplitude distortion, and eigen value spread, were controlled by W.

The received signal is processed by a linear, 11-tap FIR equalizer adapted with the LMS algorithm

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REFERENCES

“Digital Signal Processing using MATLAB” demos by Charulatha Devi.

Georgi Illiev and Nikola Kasabov, "Channel Equalization using Adaptive Filtering with Averaging", University of Otago, Newzeland.

M Reuter, J Zedlier, "Nonlinear effects in LMS adaptive equalizers", IEEE Trans.Signal Processing, June1999.