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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
179179169132
171130
169183
164182
179180
176179
180179
178178167131
167131
165179
170179
177179
182171
177177
168179169130
165132
166187
163194
176116
15394
153183
160183
412
451
4253
4255
5153
5452
5152
5151414
432
4155
3654
5152
4851
5251
5050393
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3751
4251
4951
5443
4949
4051412
374
3859
3566
4812
2534
2555
3655
-128
412
451
4253
4255
5153
5452
5152
5151414
432
4155
3654
5152
4851
5251
5050393
393
3751
4251
4951
5443
4949
4051412
374
3859
3566
4812
2534
2555
3655
13
42
12
09
40
21
13
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
4024
1914
1610
1212
1116
Q-table
13
42
12
09
40
21
13
4430
55
47
73
30
46
32
16113
916
109
621
179
3310
810
17201024
2727
132
6078
4413
1827
2738
56313
00
00
00
00
00
00
00
0000
00
00
00
00
00
00
0000
00
00
01
10
11
01
1100
01
01
23
21
13
23
520
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
00
00
00
00
00
00
0000
00
00
00
00
00
00
0000
00
00
01
10
11
01
1100
01
01
23
21
13
23
520
(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
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