Basic System Properties Short notes | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


Basic S ystem Properties in Digital Signal Processing
1. Linearity
A system is linear if it satisfies the principles of superposition and homogeneity . F or inputs x
1
[n]
andx
2
[n] , and scalarsa andb , a linear systemT produces:
T{ax
1
[n]+bx
2
[n]} =aT{x
1
[n]}+bT{x
2
[n]}
This implies the system’ s response to a weighted sum of inputs is the same weighted sum of
individual responses.
2. Time-Invariance
A system is time-invariant if a time shift in the input results in an identical time shift in the
output. F or an inputx[n] and outputy[n] =T{x[n]} , if the input is shifted b yn
0
, the output is:
T{x[n-n
0
]} =y[n-n
0
]
This property ensures the system’ s behavior does not change over time.
3. Causality
A system is causal if the output at an y time n depends only on the current and past inputs
(x[n],x[n-1],... ). F or a system to be causal:
y[n] =T{x[m],m=n}
Non-causal systems depend on future inputs, which is impr actical for real-time processing.
4. Stability
A system is stable (in the bounded-input, bounded-output sense) if every bounded input pro-
duces a bounded output. If|x[n]|=M
x
<8 for alln , then for a stable system:
|y[n]|=M
y
<8
This ensures the system does not produce unbounded outputs for finite inputs.
5. Memory
A system has memory if its output depends on past or future inputs in addition to the current
input. A memoryless system’ s output depends only on the current input:
y[n] =T{x[n]}
S ystems with memory , such as filters, use past inputs to compute the output.
1