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
댓글