1. Lect2 Time Domain Analysis

2. 2.1 Introduction
a) Convolution:- allows us to find the output signal from any LTI processor in response to any input.
b) Impulse response:- this is the response of the processor to the unit impulse response δ(n).

3. 2.2 Describing Digital Signals with Impulse Function
The digital signal x[n] is shown in fig(2.1) and it is clear that x[n] may be considered as the superposition or summation of the more basic impulse signals shown in parts

4. 2.3 Describing Digital LTI Processors
2.3.1 The Impulse Response

5. 2.3.2The Step Response
The unit step function u(n) is the running sum of the unit impulse δ[n],so the step response of a LTI processor is the running sum of its impulse response.

6. 2.4. correlation and convolution
Cross-Correlation function is a measure of similarities between two signals.
Auto- Correlation is a special case from the cross correlation. Auto correlation represents the cross correlation of the function with itself.
Convolution represents the relation between the input and the output of linear system or convolution represents how the input to a system interacts with the system to produce the output.

7. 2.4. 1 convolution Description
The relationship between the input to a linear shift invariant system, x[n] and the output y[n] is given by the convolution sum according the following equation.
1) Circular convolution
2) Linear convolution

9. 2.4.2 Performing Convolution
a) Direct Evaluation
b) Decomposition of x[n] into a set of weighted shifted impulse method:
1. Plot both x[n] and h[n] as function of n.
2. x[n] is decomposed into a set of weighted, shifted impulses.
3. Generate for each weighted shifted impulses (mentioned in step 2) its own version of the system's impulse response
4. The output y[n] is found by superposition of all these individual response.
c) Slide Rule Method
Slide rule method is convenient when both x[n] and h[n] are finite in length and short in duration.

10. The Convolution Integral
If the input consists of a continuous of impulses the convolution sum may be replaced by integral and become

11. 2.5 Difference equation
The general form of a difference equation with three recursive and three nonrecursiveterms is given by