TOPICS IN DSP( Polyphase

decomposition,DCT,Gibbs phenomenon,Oversampled ADC )

PRESENTED BYMUHAMMAD YOUNAS

ROLL NO: SE-18Department of Electrical Engineering

PIEAS

POLYPHASE DECOMPOSITION

Polyphase representation permits great simplification of theoretical results and also leads to computational efficient implementation of decimation/interpolation filters.

Let

M-Branch Polyphase decomposition

El(z) is called the Polyphase component of H(z)

ZL Mh[n] el[n]

Schematic of the Diagram Showing relation ship between h[n] and Lth Polyphase component.

Polyphase Representation for FIR and IIR Filters

Efficient Structure for Fractional Decimation.

Fractional Decimator (Decimates by rational factor M/L)

Polyphase Implementation of the fractional Decimator

COMPUTATION COMPARISON

Implementation without Polyphase: Half of the input samples being processed are zeros and only

one out of M output samples is being retained. If N is the order of the filter then 2N Multiplications and 2N-1

additions are performed and only 3rd sample is being retained at the output.

Implementation with Polyphase:

N/3 is the order of each filter. so (2/3)*(N/3) multiplications and (2/3)*(N/3-1) additions performed by each filter.

Total multiplications are (2N)/3 and additions 2*(N/3-1)+2

Improved, Low Complexity Noise Cancellation Technique for Speech Signals

In speech applications, slow convergence and high computational burden are the main problems incorporating with conventional noise cancellation method.

In applications such as the elimination of background noise from speech signals, a very long filter length required due to the requirement to model very long acoustic path impulse response.

A technique used to overcome the above problems is split the signal into subbands and adapt each subband, using separate adaptive filter.So that the order of the filter is reduced.

The computational power is greatly reduced by Polyphase implementation of the filter and the noble identities.

Improved, Low Complexity Noise Cancellation Technique for Speech Signals

By Ali O. Abid Noor, Salina Abdul Samad and Aini Hussain Department of Electrical, Electronic and System Engineering, Faculty of Engineering and Built Environment, Universiti Kebangsaan Malaysia, UKM, 43600 Bangi, Selangor, Malaysia

Paper details

OVERSAMPLING ADC

Oversampling mean to sample the signal at a rate much greater than Nyquist rate (which is to sample the signal at twice the maximum frequency contained in the signal)

The ratio between actual sampling rate and Nyquist rate is called the oversampling ratio.

Oversampling ratio=M= Fs/2 fmax

BENEFITS OF OVERSAMPLING

Simplification of Anti-aliasing Filter

Reduction in ADC noise floor by spreading the quantization noise over a wider bandwidth. This makes it possible to use an ADC with fewer bits to achieve the same SNR performance as a high resolution ADC.

Quantization error in the coarse digital output can be removed by the digital decimation filter

OVERSAMPLED ADC ARCHITECTURE

Assumptions

xa(t) is zero mean wide sense stationary random process with power spectral density and autocorrelation function

xa(t) is bandlimited to

e[n] is assumed to be uniformly distributed white noise sequence with zero mean value and variance of

Signal power component remains same from input to output

Cont…

Power in x[n] is independent of oversampling ratio (M).

As the oversampling ratio M increases, Less of the quantization noise overlaps with the signal spectrum it is this effect of the oversampling that let us to improve the SNR .

Increasing M improves SNR

The low pass filter removes the quantization noise in the band pi/M <w <pi while signal component is unaltered.

Noise power at output of LPF

For a Fixed quantization noise power Pde

Doubling the sampling ratio we need ½ bit less to achieve a given signal to quantization noise ratio.

GIBBS PHENOMENON

Uniform convergence cannot be achieved at the point of discontinuity because there always persists an overshoot even when infinite terms of Fourier series are utilized.

Since Fourier series represents continuous time periodic signals in terms of complex exponentials (continuous functions)therfore it does not seem to be possible to reconstruct a discontinuous function from set of continuous one. In fact it is not. That is the reconstruction is the same as original signal except at discontinuities.

Explanation

Find the Fourier series for square wave (finite discontinuities) and then Try to reconstruct it from its coefficients.

The more coefficients are used the more signal resemble the original.

At discontinuities we see ripples that do not subside ,using more coefficient makes them narrower, but do not shorten.

AT infinite terms the ripples will not vanish however there width goes to zero which means that they posses zero energy.

Now we can assert that the reconstruction is exactly the original except at the point of discontinuity.

The extraneous peaks in the square wave's Fourier series never disappear; they are termed Gibb's phenomenon after the American physicist Willard Gibbs. They will always occur whenever the signal is discontinuous.

Fourier series approximation of square wave

Discrete Cosine

Transform.

DCT corresponds to forming a periodic and symmetric sequence from a finite length sequence in such away that original signal can be uniquely recovered.

In the same family as the Fourier Transform Converts data to frequency domain.

Represents data via summation of variable frequency cosine waves.

Since it is a discrete version, conducive to problems formatted for computer analysis.

Captures only real components of the function. Discrete Sine Transform (DST) captures odd

(imaginary) components → not as useful. Discrete Fourier Transform (DFT) captures both

odd and even components → computationally intense.

DCT-I and DCT-II

DCT-I is one dimensional transform. Any finite N length sequence is first modified at the endpoints

and then extended to have period 2N-2.The resulting sequence is

Cont…

The resulting periodic sequence x1~[n] has even periodic

symmetry about points n=0 and n=N-1,2N-2 etc which is referred to as Type 1 periodic symmetry. DCTI is defined by the Transform pair

DCTII

DCTII is two dimensional Transform. X[n] is extended to have period 2N and the periodic sequence is

Type II symmetry ,The periodic sequence x2~[n] has even

periodic symmetry about “half sample” points -1/2,N-1/2,2N-1/2 etc.The DCT-II is defined by the Transform pair

Steps in Compression (Application of DCT-II)

1. Divide the file into 8 X 8 blocks.

2. Transform the pixel information from the spatial domain to the frequency domain with the Discrete Cosine Transform.

3. Quantize the resulting values by dividing each coefficient by an integer value and rounding off to the nearest integer.

4. Look at the resulting coefficients in a zigzag order. Follow by run length and Huffman coding.

DCT on 8x8 blocks

64 pixels

64 pixels

8 pixels

8 pixels

•We will break the image into non-overlapping 8x8 blocks.

•For each block u(m,n), we will take an 8x8 DCT

2-D DCT USED IN IMAGE COMPRESSION

1

0

1

0, 2

12cos

2

12cos,

N

j

N

iji N

lj

N

kixlklkC

• The 2-D DCT is performed as two sequential 1-D DCTs

Image y: 1-D DCT x: 1-D DCT 2-D DCT

The spatial domain shows the amplitude of the color as you move through space

The frequency domain shows how quickly the amplitude of the color is changing from one pixel to the next in an image file.

Step 2:Transform

• DC level shifting

• 2D DCT

169130

173129

170181

170183

179181

182180

179180

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169183

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166187

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176116

15394

153183

160183

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5152

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432

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3654

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4851

5251

5050393

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3751

4251

4951

5443

4949

4051412

374

3859

3566

4812

2534

2555

3655

-128

412

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4253

4255

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432

4155

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4851

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40

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4430

55

47

73

30

46

32

16113

916

109

621

179

3310

810

17201024

2727

132

6078

4413

1827

2738

56313

DCT

Step 2: Quantization

99103

101120

100112

121103

9895

8778

9272

644992113

77103

10481

10968

6455

5637

3524

22186280

5669

8751

5740

2922

2416

1714

13145560

6151

5826

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1914

1610

1212

1116

Q-table

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21

13

4430

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47

73

30

46

32

16113

916

109

621

179

3310

810

17201024

2727

132

6078

4413

1827

2738

56313

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Q

the upper-left corner coefficient is called the DC coefficient, which is a measure of the average of the energy of the blockOther coefficients are called AC coefficients, coefficients correspond to high frequencies tend to be zero or near zero for most natural images

DC coefficient

Why Quantization???

To achieve further compression by representing DCT coefficients with no greater precision than is necessary to achieve the desired image quality

Generally, the “high frequency coefficients” has larger quantization values

Quantization makes most coefficients to be zero, it makes the compression system efficient, but it’s the main source that make the system “lossy”

Frequency distribution

The human eye is not very sensitive to high frequency changes – especially in photographic images, so the high frequency data can, to some extent, be discarded.

Step 3: Zigzag Scanning

Zigzag Scan

00

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(20,5,-3,-1,-2,-3,1,1,-1,-1,0,0,1,2,3,-2,1,1,0,0,0,0,0,0,1,1,0,1,EOB)

Zigzag Scan

End Of the Block:All following coefficients are zero which is followed by run length and Huffman encoding