1. Signals, System Properties

Signal: A function of one or more variables

 

Unit Impulse: \( \delta[n] = (n == 0)? 1: 0 \)

 

Unit Step: \( u[n] = (n >= 0)? 1 : 0 \)

 

\( \delta[n] = u[n] - u[n - 1] \)

 

\( u[n] = \sum_{k = 0}^\infty \delta[n - k] \)

 

 

System: A process by which input signals are transformed to output signals

 

\( x[n] \) -> Discrete-Time System -> \( y[n] \)

 

\( x(t) \) -> Continuous-Time System -> \( y(t) \)

 

 

A system is called memoryless if its output at a given time depends on the input only at that time.

( \( x(t) \) -> \( y(t) \) )

 

 

A system is called stable if all bounded inputs generate bounded outputs.

 

\( |y(t)| < B \)  if \( |x(t)| < A, |A| < \infty, |B| < \infty \)

 

 

A system is called linear if it satisfies these two conditions:

 

1. Scaling: \( x(t) \) -> \( y(t) \) => \( a x(t) \) -> \( a y(t) \)

 

2. Superposition: \( x_1(t) \) -> \( y_1(t), x_2(t) \) -> \( y_2(t) \) => \( x_1(t) + x_2(t) \) => \( y_1(t) + y_2(t) \)

 

 

A system is called time-invariant if a time shift in the input results is an identical time shift in the output

 

\( x(t) \) -> \( y(t) \) => \( x(t - T) \) -> \( y(t - T) \)

 

 

Systems that are both linear and time-invariant, referred to as LTI (Linear Time-Invariant) systems

 

\( \delta[n] \) -> Discrete Time Linear Time-Invariant System -> \( h[n] \) (impulse response)

 

\( x[n] = ... + x[-1] \delta[n + 1] + x[0] \delta[n] + x[1] \delta[n - 1] + ... \)

 

\( y[n] = ... + x[-1] h[n + 1] + x[0] h[n] + x[1] h[n - 1] + ... \)

 

For input \( x[n] \), the output is \( y[n] = \sum^\infty_{k = -\infty} x[k] h[n - k] \)

 

Convolution of \( x \) and \( h \), \( (x * h)[n] = \sum^\infty_{k = -\infty} x[k] h[n - k] \)

 

=> \( x[n] \) -> DT LTI System -> \( y[n] = (x * h)[n] \)

 

 

reference

 

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