Unknown Time Variant System

FIR Filter Model

zz+

+

Adaptive algorithm

1.Adaptive Filtering

Adaptive filters are used in variety of applications where the statistical knowledge of the signals to be filtered/ analysed are not known apriori or the signals may be slowly time variant.

We can use both FIR and IIR filters in adaptive filtering, but FIR filters are mostly used, because i) Simple filter ii) have only adjustable zeros.

In adaptive filtering the adjustable filter parameters are to be optimized.

1.1.System Modeling

Consider an unknown system to be identified. The model is shown below. The system is modeled with M adjustable coefficients.

Noise w(n)

d(n) y(n)

x (n) + -

y (n)

e(n)

error signal

The unknown system and FIR filter is excited by the input x(n) , the output of the dynamic

system is y(n) and FIR filter output is y (n) .

y (n)=∑k=0

M−1

h (k ) x (n−k )

The error signal is given by

e (n )= y (n )− y (n)

The error signal is to be minimized using mean square error criteria,

Emin=∑n=0

∞ [ y (n )−∑k=0

M−1

h (k ) x (n−K )]2

and select the filter coefficient.

1.2 Adaptive Channel Equalization

Block diagram of digital Communication system in which an adaptive equalizer is used to compensate the distortion caused by transmitting medium (channel).

a(n) s(t) s(n)

Data Samplersequence

Noise w(n)

a (n) a (n)

d(n) + -

e(n)

Error signal

In the transmitting medium the distortion is caused by ISI and thermal noise.

The output of the receiving filter is

s ( t )=∑k

ak p(t−kT s¿)¿

Transmitter (filter)

Channel(Time variant

filter)

Receiver (Filter)

Adaptive Equalizer

Decision Device

Reference Signal

Adaptive Algorithm

Ts – signaling interval durationp(t) – impulse response of the cascade connection of the filters.The sampled version of s(t) is ,

s (k )=∑n

an p(k−n)

s (k )=ak p (0 )+∑n

n≠ k

p(k−n)

In the above equation the first term represents the desired symbol and remaining term represents inter symbol interference. To avoid ISI , transmitter and receiver filter should be properly designed based on the channel characteristics.

Since channel has random characteristics, filters are designed based on the average characteristics. But this may not reduce the ISI for the larger extent.

Adaptive equalizer is used to reduce the ISI .It compensates the channel distortion so that the detected signal will be reliable.

Adaptive equalization process is done in two stepsi) Training modeii) Tracking mode.

Training Mode A known test signal (PN sequence) is transmitted. The received signal is compared with test signal at the receiver , the resultant error signal

gives the information about the channel. This error signal is used to adjust the equalizer coefficients

Tracking Mode After Training process is done, the equalizer coefficient continuously adjusted in

decision direct mode. The output of the equalizer is sent to the decision device to get the estimate. This estimate

is used to adjust the filter coefficient.

1.3 Adaptive Line Enhancer The adaptive line enhancer used to suppress wide band noise components and only

allows the narrow band signals with less attenuation. An adaptive line enhancer (ALE) is based on the straightforward concept of linear

prediction. A nearly-periodic signal can be perfectly predicted using linear combinations of its past samples, whereas a non-periodic signal cannot.

The delay, Δ, is long enough to decorrelate the broadband “noise-like” signal, resulting in a filter which extracts the narrowband periodic signal at filter output y(k) (or removes the periodic noise from a wideband signal at e(k) ).

1.4 Adaptive Noise Cancellation

The above figure illustrates the basic principles of adaptive noise canceling. The input to the adaptive filter is a noise signal w1 (n) that is highly correlated with the additive disturbance, w(n), but is uncorrelated with the clean signal s(n).

The reference signal w1(n) is filtered to produce the output w (n) that is an

estimate of the additive noise w(n). This output is then subtracted from the noisy signal x(n) to produce the system output z(n). The system output is used to control the adaptive filter and is an estimate of s(n).

1.5 Echo Cancellation: Consider a two wire and four wire transmissions in the telephone

connections. Echo is generated at hybrid which connects a 4 to 2 –wire connection. Assume that the call is made using satellite. The satellite communication

has270 ms delay.

When A speaks to B , the speech signal takes the upper transmission and lower transmission path. Then the received signal has a delay of 540 ms.

The echo cancellation is done by finding an estimate of echo and subtracting the echo from the received signal.

Receiver B

Adaptive Algorithm

Echo Canceller

Transmitter B

Receiver A

Adaptive Algorithm

Echo Canceller

Transmitter A

HybridHybrid Hybrid B

Hybrid A

The return signal is

y (n )=∑k=0

∞

h (k ) x (n−k )+v (n)

Where x(n) is speech of speaker Av(n) is the speech of speaker B + noiseh(k) is impulse response of echo path.

The estimation of echo is

y (n)=∑k=0

∞

h (k ) x (n−k )

Where h (k ) is estimate of the impulse response of the echo path.

The error signal is

e (n )= y (n )− y (n)

By adaptively controlling h(k), after some iterations, the echo effect can be minimized.

An echo signal at terminal caused by hybrid A is a near end echo and an echo signal at terminal B caused by hybrid A is a far end echo. Bothe these echoes are removed by echo cancellers.

The receiver signal at Modem A isSRA ( t )=A1 SB ( t )+ A2 S A (t−d1 )+ A3 SA ( t−d2 )

Where SB ( t )- the desired signal which is to be demodulated at modem ASA (t−d1 )- near-end echo due to hybrid ASA (t−d2)- far -end echo due to hybrid B

A1,A2,A3 –Amplitude of signal components

r A ( t )=SRA (t )+w(t )r A (t ) - Corrupted received signal due to additive noise w (t)

Let h(n) is the impulse response of the adaptive its output signal is

SA (n )=∑k=0

M −1

h ( k ) a(n−k )

Different configuration of echo cancellers are i) Symbol rate echo cancellerii) Nyquist rate echo canceller

1.5.1 Symbol rate Echo canceller

Input data a(n)

1.5.2Nyquist rate echo canceller

Transmitter Filter

Hybrid

Receiver Filter

Symbol Rate Sampler

Decision Device

Adaptive Algorithm

Echo Canceller

Input data a(n)

2. Musical Signal Processing Almost all musical programs are produced in basically two stages. First,

sound from each individual instrument is recorded in an acoustically inert studio on a single track of a multi track tape recorder. Then, the signals

Transmitter Filter

Hybrid

Receiver Filter

Nyquist Rate Sampler

Decision Device

Adaptive Algorithm

Echo Canceller

from each track are manipulated by the sound engineer to add special audio effects and are combined in a mix-down system to finally generate the stereo recording on a two-track tape recorder.

2.1 Generation of Echo effects:

x(n) y(n)

α

Echoes are simply generated by delay units. For example the direct sound and echo appearing after R periods are generated by FIR filter which is described by the difference equation,

y (n )=x (n )+αx(n−R)

Or equivalently, by the transfer function

H ( z )=1+α z−R

This single echo filter also called as comb filter.

To generate a fixed number of multiple echoes spaced R sampling periods apart with exponentially decaying amplitudes, one can use an FIR filter with a transfer function

H ( z )=1+α z−R+α 2 z−2 R+…+α N−1 z−(N −1)R+α N z−NR

H ( z )=1−α N z−NR

1−α z−R

Z-R

To generate a infinite number of echoes spaced R sampling periods apart with exponentially decaying amplitudes, one can use an IIR filter with a transfer function

2.2 Generation Reverberation The sound reaching the listener in a closed space, such as a concert hall,

consists of several components: direct sound, early reflections, and reverberation. The early reflections are composed of several closely spaced echoes that are basically delayed and attenuated copies of the direct sound, whereas the reverberation is composed of densely packed echoes reflections.

The IIR comb filter itself does not provide natural-sounding reverberations for two reasons.

First, its magnitude response is not constant for all frequencies, resulting in a “coloration” of many musical sounds that are often unpleasant for listening purposes.

Second, the output echo density, given by the number of echoes per second, generated by a unit impulse at the input, is much lower than that observed in a real room, thus causing a “fluttering” of the composite

sound. It has been observed that approximately 1000 echoes per second are necessary to create a reverberation that sounds free of flutter.

To develop a more realistic reverberation, a reverberator with an all pass structure has been proposed. Its transfer function is given by

H ( z )= α+Z−R

1+α Z−R|α|<1

2.3Generation of Chorus effects The chorus effect is achieved when several musicians are playing the

same musical piece at the same time but with small changes in the amplitudes and small timing differences between their sounds. Such an effect can also be created synthetically by a chorus generator from the music of a single musician. A simple modification of the digital filter that can be employed to simulate this sound effect.

The phasing effect is produced by processing the signal through a narrowband notch filter with variable notch characteristics and adding a scaled portion of the notch filter output to the original signal.

2.4 Generation of flanging effect There are a number of special sound effects that are often used in the

mix-down process. One such effect is called flanging. Originally, it was created by feeding the same musical piece to two tape

recorders and then combining their delayed outputs while varying the difference ∆t between their delay times. One way of varying ∆t is to slow down one of the tape recorders by placing the operator’s thumb on the flange of the feed reel, which led to the name flanging.

The FIR comb filter can be modified to create the flanging effect

3. Image Enhancement The goal of image enhancement is to improve the image quality so that

the processed image is better than the original image for a specific application or set of objectives.

3.1 Spatial domain techniques

These techniques are based on gray level mappings, where the type of mapping used depends on the criterion chosen for enhancement.

As an eg. consider the problem of enhancing the contrast of an image. Let r and s denote any gray level in the original and enhanced image respectively.

Suppose that for every pixel with level r in original image we create a pixel in the enhanced image with level S=T(r). If T(r) has the form as shown

Figure( 5.1)

The effect of this transformation will be to produce an image of higher contrast than the original by darkening the levels below a value m and brightening the levels above m in the original pixel spectrum. The technique is referred as contrast stretching.

The values of r below m are compressed by the transformation function into a narrow range of S towards the dark end of the spectrum; the opposite effect takes place for values of r above m .

In the limiting case shown in figure, T(r) produces a 2-level (binary) image. This is also referred to as image thresholding. Many powerful enhancement processing techniques can be formulated in the spatial domain of an image.

3.2 Image Enhancement by histogram modification

The histogram of an image represents the relative frequency of occurrence of the various gray levels in the image. It provides a total description of the appearance of an image. The type and degree of enhancement obtained depends on the nature of the specified histogram.

Let the variable r represent the grey level of the pixels in the image to be enhanced. Assume that the pixel values are normalized to lie in the range

0 ≤ r ≤1 with r=0 represents black

T (r ) represents white in the gray scale

For any r in [ 0,1 ], we consider transformations of the form which produce a level S for every pixel value r in the original image. It is assumed that the transformation function satisfies the conditions:

(1) T (r ) is singled valued and monotonically increasing in the interval { };

(2)0 ≤ T (r ) ≤1 for 0 ≤ r≤ 1

Condition (1) transformation preserves the order from black to white in the gray scale

Condition (2) transformation guarantees a mapping that is consistent with the allowed range of pixel values. Example of such a transformation is:

The inverse transformr=T−1 ( s) for 0≤ s≤1 where it is assumed satisfies conditions (1) or (2) wrt variables. The gray levels in an image are random quantities in the interval [0,1] . Assuming that they are continuous variables the original and transformed gray levels can be characterized by their probability density functions Pr(r) and Ps(s ). The general characteristics of an image from the density functions of its gray levels.

Figure 5.3a

This function implies that the image will have dark characters since majority of levels are concentrated inthe dark region of gray scale

Figure 5.3b

The image will have predominant light tones since majority of pixels are light gray.

It follows from elementary probability theory that if Pr(r) and T (r ) are known

and satisfies condition (1), then

The enhancement techniques are based on modifying the appearance of an image by controlling the probability density function of its gray levels via the transformation function T (r ).

4. Speech Compression A voice signal has a frequency range of 300 to 3000 Hz. It is sampled at a

rate of 8KHz and the word length of digitized signal is 12 bits. Speech compression and coding are used to reduce the redundancy present

in the voice signals. The different voice coding techniques are

a. Wave form coding – non-uniform, differential, adaptive quantization.

b. Transform coding – transform the voice signal to an orthogonal signal and then coding the transform.

c. Frequency band of coding- frequency range of voice signals are divide into discrete channels and each channel is coded separately

d. Parametric method –linear prediction.

4.1 Channel Vocoders

The channel vocoder is an analysis synthesis system. A filter bank is used to separate the bands. There are 8 to 10 filters.

The amplitude of filters are encode using level detectors and coders. Pitch and voicing information are also sent along with them . A wideband excited signal is generated at the receiving end using the

transmission pitch and voicing information. For a voiced signal , the excitation consists of a periodic signal with

appropriate frequency. For unvoiced signal, the excitation is a white noise. At the receiver end , a matching filter bank is available, so that the output

level matches the encoded value. The individual outputs are combined to produce the speech signal.

4.2 Subband Coding Subband coding is a method where the speech signal is sub dived into

several frequency bands and each band is digitally encoded separately. Let us assume the speech signal is sampled at a rate Fs samples per second.

The First frequency subdivision splits the signal spectrum into two equal

width segments a lowpass signal (0 ≤ F ≤F s

4¿ and a high pass signal

(F s

4≤ F ≤

F s

2).

The second frequency subdivision splits the lowpass signal from the first

stage into two equal sub bands a lowpass signal (0 ≤ F ≤F s

8¿ and a high pass

signal (F s

8≤ F ≤

F s

4).

Finally the third frequency subdivision splits the lowpass signals from the second stage into two equal bandwidth signals. Thus the signal is divided into four frequency bands covering three Octaves.

Decimation by a factor 2 is performed after frequency sub division. By allowing different number of bits per sample to the signals in four sub bands, we can achieve the reduction in the bit rate of digitized signal.

In the synthesis method for the subband encoded speech signals is basically the reverse process of encoding process.

The signal is adjacent lowpass and highpass frequency bands are interpolated ,filtered and combined.