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SPEECH PROCESSING ,IMAGE processing, RADAR SIGNAL processing.

Speech processing ,Image processing, Radar signal processing.

1. SATURATION ARITHMETIC and
2. SCALING

1. ZERO LIMIT CYCLE BEHAVIOR
2. Over FLOW limit cycle behavior

ROUNDING a number to B bits is ACCOMPLISHED by choosing a ROUNDED result as the b bit number closest number being unrounded.

Rounding a number to b bits is accomplished by choosing a rounded result as the b bit number closest number being unrounded.

TRUNCATION is a process of DISCARDING all bits LESS significant than LSB that is retained

Truncation is a process of discarding all bits less significant than LSB that is retained

TRUNCATION and ROUNDING

Truncation and Rounding

The product quantization errors ARISE at the out PUT of the multiplier. Multiplication of a B bit data with a b bit coefficient results a product having 2b BITS. Since a b bit REGISTER is used the multiplier output will be rounded or truncated to b bits which produces the error.

The product quantization errors arise at the out put of the multiplier. Multiplication of a b bit data with a b bit coefficient results a product having 2b bits. Since a b bit register is used the multiplier output will be rounded or truncated to b bits which produces the error.

The filter coefficients are computed to infinite precision in THEORY. But in DIGITAL computation the filter coefficients are represented in binary and are stored in registers. If a b bit register is used the filter coefficients must be rounded or TRUNCATED to b bits ,which PRODUCES an error.

The filter coefficients are computed to infinite precision in theory. But in digital computation the filter coefficients are represented in binary and are stored in registers. If a b bit register is used the filter coefficients must be rounded or truncated to b bits ,which produces an error.

1. INPUT QUANTIZATION ERRORS
2. COEFFICIENT quantization errors
3. PRODUCT quantization errors

1. LARGE DYNAMIC RANGE
2. OVERFLOW is UNLIKELY.

In floating point FORM the positive NUMBER is represented as F =2CM,where is MANTISSA, is a FRACTION such that1/2<M<1and C the exponent can be either positive or negative.

In floating point form the positive number is represented as F =2CM,where is mantissa, is a fraction such that1/2<M<1and C the exponent can be either positive or negative.

In 2’s complement form the POSITIVE number is REPRESENTED as in the sign MAGNITUDE form. To obtain the negative of the positive number ,complement all the BITS of the positive number and ADD 1 to the LSB.

In 2’s complement form the positive number is represented as in the sign magnitude form. To obtain the negative of the positive number ,complement all the bits of the positive number and add 1 to the LSB.

In 1,s complement form the POSITIVE NUMBER is REPRESENTED as in the sign MAGNITUDE form. To obtain the NEGATIVE of the positive number ,complement all the bits of the positive number.

In 1,s complement form the positive number is represented as in the sign magnitude form. To obtain the negative of the positive number ,complement all the bits of the positive number.

For sign magnitude REPRESENTATION the leading binary digit is USED to REPRESENT the sign. If it is equal to 1 the number is negative, OTHERWISE it is positive.

For sign magnitude representation the leading binary digit is used to represent the sign. If it is equal to 1 the number is negative, otherwise it is positive.

Depending on the NEGATIVE numbers are REPRESENTED there are three FORMS of fixed point arithmetic. They are sign magnitude,1’s COMPLEMENT,2’s complement

Depending on the negative numbers are represented there are three forms of fixed point arithmetic. They are sign magnitude,1’s complement,2’s complement

In FIXED point NUMBER the position of a binary point is fixed. The BIT to the RIGHT represent the fractional part and those to the left is integer part.

In fixed point number the position of a binary point is fixed. The bit to the right represent the fractional part and those to the left is integer part.

There are three types of ARITHMETIC USED in digital systems. They are fixed POINT arithmetic, FLOATING point ,block floating point arithmetic.

There are three types of arithmetic used in digital systems. They are fixed point arithmetic, floating point ,block floating point arithmetic.

The transpose of a structure is DEFINED by the following operations:

• REVERSE the directions of all branches in the signal flow graph
• Interchange the input and outputs.
• Reverse the ROLES of all nodes in the flow graph.
• Summing points BECOME branching points.
• Branching points become summing points.

ACCORDING to transposition theorem if we reverse the directions of all branch transmittance and interchange the input and output in the flowgraph, the system function remains unchanged.

The transpose of a structure is defined by the following operations:

According to transposition theorem if we reverse the directions of all branch transmittance and interchange the input and output in the flowgraph, the system function remains unchanged.

A SIGNAL FLOW graph is a graphical representation of the RELATIONSHIPS between the variables of a set of LINEAR difference equations.

A signal flow graph is a graphical representation of the relationships between the variables of a set of linear difference equations.

Quantization ERRORS can be minimized if we REALIZE an LTI system in CASCADE form.

Quantization errors can be minimized if we realize an LTI system in cascade form.

ADVANTAGES:

• The bilinear transformation provides one-to-one mapping.
• Stable continuous SYSTEMS can be MAPPED into realizable, stable digital systems.
• There is no aliasing.

Disadvantage:

• The mapping is highly non-linear PRODUCING frequency, COMPRESSION at high frequencies.
• Neither the impulse response nor the phase response of the analog filter is preserved in a digital filter obtained by bilinear transformation.

Advantages:

Disadvantage:

The effect of the non-linear compression at high FREQUENCIES can be compensated. When the desired magnitude response is piece-wise CONSTANT over frequency, this compression can be compensated by INTRODUCING a suitable pre-scaling, or pre-warping the critical frequencies by USING the formula.

The effect of the non-linear compression at high frequencies can be compensated. When the desired magnitude response is piece-wise constant over frequency, this compression can be compensated by introducing a suitable pre-scaling, or pre-warping the critical frequencies by using the formula.

• The MAPPING for the BILINEAR transformation is a one-to-one mapping that is for EVERY point Z, there is exactly one corresponding point S, and vice-versa.
• The j Ω-axis MAPS on to the unit circle |z|=1,the left half of the s-plane maps to the INTERIOR of the unit circle |z|=1 and the half of the s-plane maps on to the exterior of the unit circle |z|=1.

The BILINEAR transformation is a mapping that TRANSFORMS the left half of S-plane into the unit CIRCLE in the Z-plane only once, THUS avoiding ALIASING of frequency components.

The mapping from the S-plane to the Z-plane is in bilinear transformation is

S=2/T(1-Z-1/1+Z-1)

The bilinear transformation is a mapping that transforms the left half of S-plane into the unit circle in the Z-plane only once, thus avoiding aliasing of frequency components.

The mapping from the S-plane to the Z-plane is in bilinear transformation is

S=2/T(1-Z-1/1+Z-1)

S=2/T(1-Z-1/1+Z-1)

S=2/T(1-Z-1/1+Z-1)

In this method of digitizing an analog FILTER, the IMPULSE response of resulting digital filter is a sampled version of the impulse response of the analog filter.The transfer function of analog filter in PARTIAL FRACTION FORM.

In this method of digitizing an analog filter, the impulse response of resulting digital filter is a sampled version of the impulse response of the analog filter.The transfer function of analog filter in partial fraction form.

The mapping PROCEDURE between S-plane & Z-plane in the method of mapping of differentials is given by

H(Z) =H(S)|S=(1-Z-1)/T

The above mapping has the following characteristics:

• The LEFT half of S-plane maps inside a circle of radius ½ centered at Z= ½ in the Z-plane.
• The RIGHT half of S-plane maps into the region outside the circle of radius ½ in the Z-plane.
• The J Ω-axis maps onto the perimeter of the circle of radius ½ in the Z-plane.

The mapping procedure between S-plane & Z-plane in the method of mapping of differentials is given by

H(Z) =H(S)|S=(1-Z-1)/T

The above mapping has the following characteristics:

ONE of the simplest method for converting an analog filter into a DIGITAL filter is to approximate the DIFFERENTIAL equation by an equivalent difference equation.

d/dt y(t)=y(nT)-y(nT-T)/T

The above equation is called backward difference equation.

One of the simplest method for converting an analog filter into a digital filter is to approximate the differential equation by an equivalent difference equation.

d/dt y(t)=y(nT)-y(nT-T)/T

The above equation is called backward difference equation.

The TWO IMPORTANT PROCEDURES for digitizing the transfer function of an analog filter are:

• IMPULSE invariance method. 
• Bilinear transformation method.

The two important procedures for digitizing the transfer function of an analog filter are:

• Map the DESIRED DIGITAL FILTER SPECIFICATIONS into those for an EQUIVALENT analog filter.
• Derive the analog transfer function for the analog prototype.
• Transform the transfer function of the analog prototype into an equivalent digital filter transfer function.

In DIRECT form ΙΙ STRUCTURE, the number of MEMORY LOCATIONS required is LESS than that of direct form Ι structure.

In direct form ΙΙ structure, the number of memory locations required is less than that of direct form Ι structure.

IIR filters are of recursive TYPE whereby the present o/p sample depends on present i/p, PAST i/p samples and o/p samples. The design of IIR filter is realizable and stable.

The impulse response h(N) for a realizable filter is

h(n)=0 for n≤0

IIR filters are of recursive type whereby the present o/p sample depends on present i/p, past i/p samples and o/p samples. The design of IIR filter is realizable and stable.

The impulse response h(n) for a realizable filter is

h(n)=0 for n≤0

For an M_stage FILTER , αm-1(0) =1 and KM = αm(m)

αm-1(K) = αm(k) - αm(m) • αm(m-k) , 1≤k≤m-1

1-αm2 (m)

For an M_stage filter , αm-1(0) =1 and km = αm(m)

αm-1(k) = αm(k) - αm(m) • αm(m-k) , 1≤k≤m-1

1-αm2 (m)

The cascade form REALIZATION is PREFERRED when COMPLEX ZEROS with absolute magnitude is less than one.

The cascade form realization is preferred when complex zeros with absolute magnitude is less than one.

Frequency SAMPLING METHOD is attractive for narrow band frequency selective filters where only a few of the SAMPLES of the frequency response are non ZERO.

Frequency sampling method is attractive for narrow band frequency selective filters where only a few of the samples of the frequency response are non zero.

In FREQUENCY sampling method the desired magnitude response is sampled and a linear PHASE response is specified .The SAMPLES of desired frequency response are identified as DFT coefficients. The FILTER coefficients are then DETERMINED as the IDFT of this set of samples.

In frequency sampling method the desired magnitude response is sampled and a linear phase response is specified .The samples of desired frequency response are identified as DFT coefficients. The filter coefficients are then determined as the IDFT of this set of samples.

• It PROVIDES flexibility for the designer to SELECT the side lobe level and N
• It has the ATTRACTIVE property that the side lobe level can be varied continuously from the low VALUE in the Blackman window to the HIGH value in the rectangular window

The necessary and sufficient condition for linear phase characteristic in FIR filter is, the IMPULSE response h(n) of the SYSTEM should have the symmetry PROPERTY i.e.,

H(n) = h(N-1-n)

where N is the DURATION of the sequence.

The necessary and sufficient condition for linear phase characteristic in FIR filter is, the impulse response h(n) of the system should have the symmetry property i.e.,

H(n) = h(N-1-n)

where N is the duration of the sequence.

1. Rectangular WINDOW
2. Hamming window
3. Hanning window
4. Bartlett window
5. Kaiser window

1. Rectangular window:

The equation for Rectangular window is given by

W(n)= 1 0 ≤ n ≤ M-1

0 otherwise

2. Hamming window:

The equation for Hamming window is given by

WH(n)= 0.54-0.46 cos 2пn/M-1 0 ≤ n ≤ M-1

0 otherwise 

3. Hanning window:

The equation for Hanning window is given by

WHn(n)= 0.5[1- cos 2пn/M-1 ] 0 ≤ n ≤ M-1

0 otherwise

4. Bartlett window:

The equation for Bartlett window is given by

WT(n)= 1-2|n-(M-1)/2| 0 ≤ n ≤ M-1

M-1

0 otherwise

5. Kaiser window:

The equation for Kaiser window is given by

WK(n)= Io[α√1-( 2n/N-1)2] for |n| ≤ N-1

Io(α) 2

0 otherwise

where α is an INDEPENDENT PARAMETER.

1. Rectangular window:

The equation for Rectangular window is given by

W(n)= 1 0 ≤ n ≤ M-1

0 otherwise

2. Hamming window:

The equation for Hamming window is given by

WH(n)= 0.54-0.46 cos 2пn/M-1 0 ≤ n ≤ M-1

0 otherwise 

3. Hanning window:

The equation for Hanning window is given by

WHn(n)= 0.5[1- cos 2пn/M-1 ] 0 ≤ n ≤ M-1

0 otherwise

4. Bartlett window:

The equation for Bartlett window is given by

WT(n)= 1-2|n-(M-1)/2| 0 ≤ n ≤ M-1

M-1

0 otherwise

5. Kaiser window:

The equation for Kaiser window is given by

Wk(n)= Io[α√1-( 2n/N-1)2] for |n| ≤ N-1

Io(α) 2

0 otherwise

where α is an independent parameter.

The desirable CHARACTERISTICS of the window are:

1. The central lobe of the frequency response of the window should contain most of the energy and should be narrow.
2. The highest SIDE lobe level of the frequency response should be small.
3. The side lobes of the frequency response should decrease in energy rapidly as ω TENDS to п.

The desirable characteristics of the window are:

One possible way of finding an FIR FILTER that approximates H(EJW) would be to truncate the infinite Fourier series at n=±(N-1/2).Direct truncation of the series will lead to fixed percentage overshoots and undershoots before and after an approximated discontinuity in the frequency response.

One possible way of finding an FIR filter that approximates H(ejw) would be to truncate the infinite Fourier series at n=±(N-1/2).Direct truncation of the series will lead to fixed percentage overshoots and undershoots before and after an approximated discontinuity in the frequency response.

There are THREE well known methods for DESIGNING FIR filters with LINEAR phase .They are (1.)WINDOW method (2.)Frequency sampling method (3.)Optimal or minimax design.

There are three well known methods for designing FIR filters with linear phase .They are (1.)Window method (2.)Frequency sampling method (3.)Optimal or minimax design.

FIR filter:

• IR filter

• These filters can be easily designed to have perfectly linear phase.
• FIR filters can be realized recursively and non-recursively.
• Greater flexibility to control the shape of their magnitude RESPONSE.
• ERRORS due to ROUND off NOISE are less severe in FIR filters, mainly because feedback is not used.

IIR filter:

• These filters do not have linear phase.
• IIR filters are easily realized recursively.
• Less flexibility, usually limited to specific kind of filters.
• The round off noise in IIR filters is more.

FIR filter:

IIR filter:

BASED on frequency response the FILTERS can be classified as:

1. Lowpass FILTER
2. Highpass filter
3. Bandpass filter
4. Bandreject filter

Based on frequency response the filters can be classified as:

Based on IMPULSE RESPONSE the filters are of two types:

1. IIR filter
2. FIR filter

The IIR filters are of recursive type, whereby the present output sample DEPENDS on the present INPUT, past input samples and output samples.

The FIR filters are of non recursive type, whereby the present output sample depends on the present input sample and previous input samples.

Based on impulse response the filters are of two types:

The IIR filters are of recursive type, whereby the present output sample depends on the present input, past input samples and output samples.

The FIR filters are of non recursive type, whereby the present output sample depends on the present input sample and previous input samples.

Differences: 

1. The input is bit reversed while the output is in natural order for DIT, WHEREAS for DIF the output is bit reversed while the input is in natural order.
2. The DIF butterfly is slightly different from the DIT butterfly, the difference being that the COMPLEX multiplication takes place after the add-subtract operation in DIF.

Similarities:

Both algorithms require same NUMBER of operations to compute the DFT.Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

Differences: 

Similarities:

Both algorithms require same number of operations to compute the DFT.Both algorithms can be done in place and both need to perform bit reversal at some place during the computation.

Overlap-save method:

In this method the SIZE of the input data block is N=L+M-1

Each data block consists of the last M-1 data points of the previous data block followed by L NEW data points

In each output block M-1 points are corrupted due to aliasing as circular convolution is employed

To form the output sequence the FIRST

M-1 data points are discarded in each output block and the remaining data are FITTED together

Overlap-add method:

In this method the size of the input data block is L

Each data block is L points and we append M-1 zeros to compute N point DFT

In this no corruption due to aliasing as linear convolution is performed using circular convolution

To form the output sequence the last

M-1 points from each output block is added to the first M-1 points of the succeeding block

Overlap-save method:

In this method the size of the input data block is N=L+M-1

Each data block consists of the last M-1 data points of the previous data block followed by L new data points

In each output block M-1 points are corrupted due to aliasing as circular convolution is employed

To form the output sequence the first

M-1 data points are discarded in each output block and the remaining data are fitted together

Overlap-add method:

In this method the size of the input data block is L

Each data block is L points and we append M-1 zeros to compute N point DFT

In this no corruption due to aliasing as linear convolution is performed using circular convolution

To form the output sequence the last

M-1 points from each output block is added to the first M-1 points of the succeeding block

Linear CONVOLUTION:

If x(n) is a sequence of L NUMBER of SAMPLES and h(n) with M number of samples, after convolution y(n) will have N=L+M-1 samples.

It can be used to FIND the response of a linear filter.

Zero padding is not NECESSARY to find the response of a linear filter.

Circular convolution:

If x(n) is a sequence of L number of samples and h(n) with M samples, after convolution y(n) will have N=max(L,M) samples.

It cannot be used to find the response of a filter.

Zero padding is necessary to find the response of a filter.

Linear convolution:

If x(n) is a sequence of L number of samples and h(n) with M number of samples, after convolution y(n) will have N=L+M-1 samples.

It can be used to find the response of a linear filter.

Zero padding is not necessary to find the response of a linear filter.

Circular convolution:

If x(n) is a sequence of L number of samples and h(n) with M samples, after convolution y(n) will have N=max(L,M) samples.

It cannot be used to find the response of a filter.

Zero padding is necessary to find the response of a filter.

Once the butterfly operation is performed on a pair of complex numbers (a,b) to PRODUCE (A,B), there is no need to save the input pair. We can store the RESULT (A,B) in the same locations as (a,b). Since the same STORAGE locations are used troughout the computation we say that the computations are done in PLACE.

Once the butterfly operation is performed on a pair of complex numbers (a,b) to produce (A,B), there is no need to save the input pair. We can store the result (A,B) in the same locations as (a,b). Since the same storage locations are used troughout the computation we say that the computations are done in place.

The APPLICATIONS of FFT ALGORITHM includes:

1. Linear filtering
2. CORRELATION
3. Spectrum ANALYSIS

The applications of FFT algorithm includes: