**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.

- Map the desired digital filter specifications into those for an equivalent analog filter.
- Derive the analog trfer function for the analog prototype.
- Trform the trfer function of the analog prototype into an equivalent digital filter trfer function.

**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

Truncation and Rounding

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

The bilinear trformation is a mapping that trforms 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 trformation is

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

**Advantages:**

- The bilinear trformation 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 trformation.

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

- Impulse invariance method.
- Bilinear trformation method.

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).

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

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)

- Direct valuation by contour integration.
- Expion into series of terms in the variable Z and Z-@
- Partial fraction expion and look up table.

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

- 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=@

- Saturation arithmetic and
- Scaling

- Input quantization errors
- Coefficient quantization errors
- Product quantization errors

- Large dynamic range
- Overflow is unlikely.

- Periodicity
- Linearity and symmetry
- Multiplication of two DFTs
- Circular convolution
- Time reversal
- Circular time shift and frequency shift
- Complex conjugate
- Circular correlation

**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.

- 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

**Based on impulse response the filters are of two types:**

- IIR filter
- 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.

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:**

- We can get better display of the frequency spectrum.
- With zero padding the DFT can be used in linear filtering.

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.

**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

**The trpose 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 trposition theorem if we reverse the directions of all branch trmittance and interchange the input and output in the flowgraph, the system function remains unchanged.

Speech processing ,Image processing, Radar signal processing.

**The applications of FFT algorithm includes:**

- Linear filtering
- Correlation
- Spectrum analysis