what are the properties of convolution sum
The convolution sum, a fundamental operation in signal processing and systems theory, possesses several important properties that are essential for understanding and analyzing signals and systems. Here are the key properties of the convolution sum:
1. Commutativity
Convolution is commutative, meaning the order of the input signals can be reversed without changing the result:
x[n]∗h[n]=h[n]∗x[n]x[n] * h[n] = h[n] * x[n]x[n]∗h[n]=h[n]∗x[n]
2. Associativity
Convolution is associative, meaning the grouping of input signals can be changed without altering the result:
(x[n]∗h[n])∗g[n]=x[n]∗(h[n]∗g[n])(x[n] * h[n]) * g[n] = x[n] * (h[n] * g[n])(x[n]∗h[n])∗g[n]=x[n]∗(h[n]∗g[n])
3. Distributivity
Convolution distributes over addition of signals:
x[n]∗(h[n]+g[n])=x[n]∗h[n]+x[n]∗g[n]x[n] * (h[n] + g[n]) = x[n] * h[n] + x[n] * g[n]x[n]∗(h[n]+g[n])=x[n]∗h[n]+x[n]∗g[n]
4. Identity Element
The identity element of convolution is the impulse signal δ[n]\delta[n]δ[n]:
x[n]∗δ[n]=x[n]x[n] * \delta[n] = x[n]x[n]∗δ[n]=x[n]
5. Shift Invariance
Convolution is shift-invariant, meaning shifting one of the signals in time results in a corresponding shift in the output signal:
x[n]∗h[n−n0]=(x[n]∗h[n])n0x[n] * h[n - n_0] = (x[n] * h[n])_{n_0}x[n]∗h[n−n0?]=(x[n]∗h[n])n0??
where (x[n]∗h[n])n0(x[n] * h[n])_{n_0}(x[n]∗h[n])n0?? denotes the convolution result shifted by n0n_0n0? units.
6. Time Reversal
Convolution exhibits time-reversal symmetry:
x[n]∗h[n]=x[−n]∗h[−n]x[n] * h[n] = x[-n] * h[-n]x[n]∗h[n]=x[−n]∗h[−n]
7. Area (Integral) Under Convolution
The area under the convolution of two signals in the time domain is equal to the product of the areas under the original signals in the frequency domain:
∫−∞∞(x(τ)∗h(t−τ)) dτ=X(f)⋅H(f)\int_{-\infty}^{\infty} (x(\tau) * h(t - \tau)) \, d\tau = X(f) \cdot H(f)∫−∞∞?(x(τ)∗h(t−τ))dτ=X(f)⋅H(f)
where X(f)X(f)X(f) and H(f)H(f)H(f) are the Fourier transforms of x(t)x(t)x(t) and h(t)h(t)h(t), respectively.
8. Zero State Response
The output of a linear time-invariant (LTI) system in response to a zero input (i.e., x[n]=0x[n] = 0x[n]=0) is zero:
0∗h[n]=00 * h[n] = 00∗h[n]=0
9. Convolution of Delta Functions
The convolution of delta functions represents the shifting property:
δ[n−n0]∗δ[n−n1]=δ[n−(n0+n1)]\delta[n - n_0] * \delta[n - n_1] = \delta[n - (n_0 + n_1)]δ[n−n0?]∗δ[n−n1?]=δ[n−(n0?+n1?)]
These properties make convolution a powerful tool for analyzing and understanding the behavior of linear systems and signals in both time and frequency domains. They also underpin many mathematical techniques and algorithms used in various fields such as signal processing, communications, and control systems.
