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It is a popular form of the FFT algorithm. In this the output SEQUENCE X(k) is DIVIDED into smaller and smaller sub-sequences , that is why the name Decimation In FREQUENCY.

It is a popular form of the FFT algorithm. In this the output sequence X(k) is divided into smaller and smaller sub-sequences , that is why the name Decimation In Frequency.

Decimation-In-Time algorithm is used to calculate the DFT of a N point sequence. The idea is to BREAK the N point sequence into two sequences, the DFTs of which can be COMBINED to give the DFt of the

original N point sequence.This algorithm is CALLED DIT because the sequence x(n) is often splitted into smaller sub- sequences.

Decimation-In-Time algorithm is used to calculate the DFT of a N point sequence. The idea is to break the N point sequence into two sequences, the DFTs of which can be combined to give the DFt of the

original N point sequence.This algorithm is called DIT because the sequence x(n) is often splitted into smaller sub- sequences.

The FFT algorithm is most EFFICIENT in CALCULATING N point DFT. If the number of output points N can be EXPRESSED as a POWER of 2 that is N=2M, where M is an integer, then this algorithm is known as radix-2 algorithm.

The FFT algorithm is most efficient in calculating N point DFT. If the number of output points N can be expressed as a power of 2 that is N=2M, where M is an integer, then this algorithm is known as radix-2 algorithm.

The number of multiplications and ADDITIONS REQUIRED to COMPUTE N point DFT using radix-2 FFT are N log2 N and N/2 log2 N respectively,.

The number of multiplications and additions required to compute N point DFT using radix-2 FFT are N log2 N and N/2 log2 N respectively,.

The Fast Fourier Transform is an algorithm USED to compute the DFT. It MAKES use of the symmetry and periodicity PROPERTIES of twiddle factor to effectively reduce the DFT COMPUTATION time.It is based on the fundamental PRINCIPLE of decomposing the computation of DFT of a sequence of length N into successively smaller DFTs.

The Fast Fourier Transform is an algorithm used to compute the DFT. It makes use of the symmetry and periodicity properties of twiddle factor to effectively reduce the DFT computation time.It is based on the fundamental principle of decomposing the computation of DFT of a sequence of length N into successively smaller DFTs.

The direct evaluation DFT requires N2 complex multiplications and N2 –N complex ADDITIONS. Thus for large VALUES of N direct evaluation of the DFT is DIFFICULT. By using FFT algorithm the number of complex COMPUTATIONS can be reduced. So we use FFT.

The direct evaluation DFT requires N2 complex multiplications and N2 –N complex additions. Thus for large values of N direct evaluation of the DFT is difficult. By using FFT algorithm the number of complex computations can be reduced. So we use FFT.

In this method the data sequence is divided into N point SECTIONS xi(n).Each section contains the last M-1 data points of the PREVIOUS section followed by L new data points to form a data sequence of length N=L+M-1.In CIRCULAR convolution of xi(n) with h(n) the first M-1 points will not AGREE with the linear convolution of xi(n) and h(n) because of aliasing, the remaining points will agree with linear convolution. Hence we discard the first (M-1) points of filtered section xi(n) N h(n). This process is repeated for all sections and the filtered sections are abutted together.

In this method the data sequence is divided into N point sections xi(n).Each section contains the last M-1 data points of the previous section followed by L new data points to form a data sequence of length N=L+M-1.In circular convolution of xi(n) with h(n) the first M-1 points will not agree with the linear convolution of xi(n) and h(n) because of aliasing, the remaining points will agree with linear convolution. Hence we discard the first (M-1) points of filtered section xi(n) N h(n). This process is repeated for all sections and the filtered sections are abutted together.

In this method the size of the INPUT data block xi(n) is L. To each data block we append M-1 ZEROS and perform N POINT circular convolution of xi(n) and H(n). Since each data block is terminated with M-1 zeros the last M-1 points from each OUTPUT block must be overlapped and added to first M-1 points of the succeeding blocks.This method is called overlap-add method.

In this method the size of the input data block xi(n) is L. To each data block we append M-1 zeros and perform N point circular convolution of xi(n) and h(n). Since each data block is terminated with M-1 zeros the last M-1 points from each output block must be overlapped and added to first M-1 points of the succeeding blocks.This method is called overlap-add method.

The two METHODS USED for the sectional convolution are:

1. The overlap-add METHOD and
2. Overlap-save method.

The two methods used for the sectional convolution are:

If the data SEQUENCE x(n) is of long DURATION it is very difficult to obtain the output sequence y(n) DUE to limited MEMORY of a digital computer. Therefore, the data sequence is divided up into smaller sections. These sections are processed separately one at a TIME and controlled later to get the output.

If the data sequence x(n) is of long duration it is very difficult to obtain the output sequence y(n) due to limited memory of a digital computer. Therefore, the data sequence is divided up into smaller sections. These sections are processed separately one at a time and controlled later to get the output.

Let the sequence x(n) has a length L. If we WANT to find the N-point DFT(N>L) of the sequence x(n), we have to add (N-L) zeros to the sequence x(n). This is known as ZERO padding.

The uses of zero padding are:

1. We can get better display of the FREQUENCY SPECTRUM.
2. With zero padding the DFT can be used in linear filtering.

Let the sequence x(n) has a length L. If we want to find the N-point DFT(N>L) of the sequence x(n), we have to add (N-L) zeros to the sequence x(n). This is known as zero padding.

The uses of zero padding are:

CONSIDER two finite duration SEQUENCES x(n) and h(n) of duration L samples and M samples. The linear convolution of these two sequences produces an OUTPUT sequence of duration L+M-1 samples, whereas , the circular convolution of x(n) and h(n) give N samples where N=max(L,M).In ORDER to obtain the number of samples in circular convolution equal to L+M-1, both x(n) and h(n) must be appended with APPROPRIATE number of zero valued samples. In other words by increasing the length of the sequences x(n) and h(n) to L+M-1 points and then circularly convolving the resulting sequences we obtain the same result as that of linear convolution.

Consider two finite duration sequences x(n) and h(n) of duration L samples and M samples. The linear convolution of these two sequences produces an output sequence of duration L+M-1 samples, whereas , the circular convolution of x(n) and h(n) give N samples where N=max(L,M).In order to obtain the number of samples in circular convolution equal to L+M-1, both x(n) and h(n) must be appended with appropriate number of zero valued samples. In other words by increasing the length of the sequences x(n) and h(n) to L+M-1 points and then circularly convolving the resulting sequences we obtain the same result as that of linear convolution.

1. PERIODICITY
2. Linearity and SYMMETRY
3. MULTIPLICATION of two DFTs
4. Circular convolution
5. Time reversal
6. Circular time shift and frequency shift
7. COMPLEX CONJUGATE
8. Circular correlation

• Direct VALUATION by contour integration.
• EXPANSION into series of terms in the variable Z and Z-1.
• PARTIAL fraction expansion and look up table.

• The ROC does not contain any POLES.
• When x(n) is of FINITE duration then ROC is ENTIRE Z-plane except Z=0 or Z=∞.
• If X(Z) is causal,then ROC INCLUDES Z=∞.
• If X(Z) is anticasual,then ROC includes Z=0.

The region of convergence (ROC) of X(Z) the SET of all values of Z for which X(Z) attain FINAL VALUE.

The region of convergence (ROC) of X(Z) the set of all values of Z for which X(Z) attain final value.

A SYSTEM is SAID to be stable if we get BOUNDED output for bounded input.

A system is said to be stable if we get bounded output for bounded input.

A system is called time invariant if its output , INPUT CHARACTERISTICS dos not change with time.

e.g.y(N)=x(n)+x(n-1)

A system is called time VARIANT if its input, output characteristics CHANGES with time.

e.g.y(n)=x(-n).

A system is called time invariant if its output , input characteristics dos not change with time.

e.g.y(n)=x(n)+x(n-1)

A system is called time variant if its input, output characteristics changes with time.

e.g.y(n)=x(-n).

A DISCRETE time system is called static or memory less if its output at any instant n depends almost on the input sample at the same time but not on PAST and future SAMPLES of the input.

e.g. y(n) =a x (n)

In anyother case the system is said to be dynamic and to have memory. 

e.g. (n) =x (n)+3 x(n-1)

A discrete time system is called static or memory less if its output at any instant n depends almost on the input sample at the same time but not on past and future samples of the input.

e.g. y(n) =a x (n)

In anyother case the system is said to be dynamic and to have memory. 

e.g. (n) =x (n)+3 x(n-1)

• Static and dynamic system.
• TIME INVARIANT and time VARIANT system.
• Causal and anticausal system.
• Linear and Non-linear system.
• Stable and Unstable system.

A real value SIGNAL x (n) is called symmetric (even) if x (-n) =x (n). On the other HAND the signal is called antisymmetric (ODD) if x (-n) =x (n).

A real value signal x (n) is called symmetric (even) if x (-n) =x (n). On the other hand the signal is called antisymmetric (odd) if x (-n) =x (n).

A signal x (N) is periodic in PERIOD N, if x (n+N) =x (n) for all n. If a signal does not satisfy this EQUATION, the signal is called APERIODIC signal.

A signal x (n) is periodic in period N, if x (n+N) =x (n) for all n. If a signal does not satisfy this equation, the signal is called aperiodic signal.

The TYPES of discrete time signals are:

• Energy and power signals
• Periodic and Aperiodic signals
• Symmetric(even) and ANTISYMMETRIC (odd) signals

The types of discrete time signals are:

Unit SAMPLE sequence (unit impulse)

δ (n)= {1 n=0

0 Otherwise

Unit step signal

U (n) ={ 1 n>=0

0 Otherwise

Unit ramp signal

Ur(n)={n for n>=0

0 Otherwise

Exponential signal

X (n)=an where a is real

x(n)-Real signal

Unit sample sequence (unit impulse)

δ (n)= {1 n=0

0 Otherwise

Unit step signal

U (n) ={ 1 n>=0

0 Otherwise

Unit ramp signal

Ur(n)={n for n>=0

0 Otherwise

Exponential signal

x (n)=an where a is real

x(n)-Real signal

A discrete or an algorithm that PERFORMS some PRESCRIBED operation on a discrete time signal is CALLED discrete time system.

A discrete or an algorithm that performs some prescribed operation on a discrete time signal is called discrete time system.

A DISCRETE time signal X (N) is a function of an independent variable that is an INTEGER. a discrete time signal is not DEFINED at instant between two successive samples.

A discrete time signal x (n) is a function of an independent variable that is an integer. a discrete time signal is not defined at instant between two successive samples.