Task 9: Convolution Countdown

• Objective: To theoretically learn basics of Digital Signal Processing
• Task: Learn basics of DSP including an introduction to signals, systems and mathematical transforms such as Z-Transform and Fourier Transform. Perform a simple Linear Convolution in MATLAB for two 4 sample discrete signals.

Signals and Systems

Signals:

A Signal is a function of one or more independent variables (like time). It conveys the information. A signal cannot be a constant like a DC signal. It is always changing. Example: f(t), g(t), x(n)= [1, 2, 3, 4] where 1, 2, 3, 4 are the amplitudes of the signal.

System:

• The meaningful interconnection of physical devices and components is called as a system. It processes the input signal to produce an output signal.
• Properties:

1. Linearity: T(ax1​[n]+bx2​[n]) = aT(x1​[n])+bT(x2​[n])
2. Time invariance: Behavior of the system with respect to time
3. Causality: Output at a time depends only on present and past inputs
4. Stability: Bounded input gives Bounder output.

Types of Signals in DSP

1. Based on Time: Continuous vs Discrete-Time

1. Continuous-Time Signal: It is defined for every value of time. Example: x(t)=sin⁡(2πt)
2. Discrete-Time Signal: Defined only at discrete time points (at integer values). Example: x[n]=sin⁡(πn)

2. Based on Determinism: Deterministic vs Random

1. Deterministic Signal: Behavior is exactly predictable. Example: x[n]=cos⁡(πn)
2. Random (Stochastic) Signal: Behavior is not exactly predictable. Example: Noise signal

3. Based on Periodicity: Periodic vs Non-periodic

1. Periodic Signal: Repeats after a fixed time period. Example: x[n]=cos⁡(2πn/4)
2. Non-periodic Signal: Does not repeat. Example: x[n]=e^(−n)u[n]

4. Based on Symmetry: Even, Odd, or Neither

• Even Signal: Symmetric about the origin. x[n]=x[−n]. Example: x[n]=n^2
• Odd Signal: Asymmetric about the origin. x[-n]=−x[n]. Example: x[-n]=n = nx[n]=n

6. Other Signal Types

1. Unit Impulse: A is equal to 1 at time 0
2. Unit Step: A is equal to 1 for time >= 0
3. Ramp: A is equal to t for t>= 0
4. Parabolic: A is equal to t^2 for t>=0

Z-Transform

The Z-Transform is used to analyze discrete-time signals and systems in the context of system behavior, stability, and frequency response. It transforms a discrete-time signal from the time domain into the z-domain, which is a complex frequency domain. It is used for studying frequency response.

Discrete Time Fourier Transform

It is used to find the magnitude and phase of the signal. It is present in 2 types, Polar and Rectangular Form. It is use din speech and image processing.

Convolution

Convolution is used to find the output of a system when the input and the system's response (called impulse response) are known. It combines the two signals to find the third signal.

Convolution of two 4 discrete signals

Matlab Code

• x = [1, 2, 3, 4];

• h = [1, 0, -1, 2];

• y = conv(x, h);

• disp('Linear Convolution Result:');

• disp(y);

• n = 0:length(y)-1;

• stem(n, y);

• title('Linear Convolution of x(n) and h(n)');

• xlabel('n');

• ylabel('y(n)');