The order of convolution does not matter. If x[n] is convolved with h[n], the result is identical to convolving h[n] with x[n].

x[n]∗h[n]=h[n]∗x[n]

Mathematical Proof (Concept)

This property can be rigorously proven by performing a simple change of the summation variable in the convolution sum formula. If we let m=n−k in the expression for x[n]∗h[n], then k=n−m. As k goes from −∞ to ∞, m also goes from ∞ to −∞ (or vice-versa, depending on how you think about the limits, but the range remains the same). Substituting these into the sum:

y[n]=∑m=∞−∞ x[n−m]h[m] Rearranging the summation (which is allowed for absolute summable sequences) and changing the dummy variable back to k:
y[n]=∑k=−∞∞ h[k]x[n−k] This is precisely the definition of h[n]∗x[n], thus proving commutativity.

Interpretation

From a system analysis perspective, this means that if you have an LTI system characterized by its impulse response h[n], and you apply an input signal x[n] to it, the resulting output y[n] is exactly the same as if you hypothetically constructed an LTI system whose impulse response was x[n] and then applied h[n] as the input to that system. While this might seem abstract, it implies a certain symmetry and flexibility in how we view the roles of input and system.

This property applies when multiple LTI systems are connected in cascade (i.e., in series). It states that the grouping of convolution operations does not affect the final result.

(x[n]∗h1 [n])∗h2 [n]=x[n]∗(h1 [n]∗h2 [n])

Interpretation

Imagine a signal x[n] passing sequentially through two LTI systems. First, system 1 (with impulse response h1 [n]) processes x[n] to produce an intermediate output. Then, this intermediate output becomes the input to system 2 (with impulse response h2 [n]), yielding the final output y[n]. The associativity property states that the overall final output y[n] is the same regardless of whether you:

• First convolve x[n] with h1 [n] to get the first intermediate signal, and then convolve that intermediate signal with h2 [n] to get y[n]. OR
• First determine the overall equivalent impulse response of the cascaded systems. The overall impulse response of two LTI systems in cascade is simply the convolution of their individual impulse responses: hoverall [n]=h1 [n]∗h2 [n]. Then, you convolve the original input x[n] with this hoverall [n] to get y[n].

Significance

This property is incredibly powerful for simplifying the analysis and design of complex systems built from interconnected LTI components. It also implies that the order in which cascaded LTI systems are connected can be swapped without altering the overall system's input-output behavior. This flexibility is often exploited in digital filter design.

This property applies when LTI systems are connected in parallel. It states that convolution distributes over addition, similar to how multiplication distributes over addition.

x[n]∗(h1 [n]+h2 [n])=(x[n]∗h1 [n])+(x[n]∗h2 [n])

Interpretation

Consider a scenario where an input signal x[n] is applied simultaneously to two separate LTI systems, h1 [n] and h2 [n], connected in parallel. The individual outputs from these two systems are then summed together to produce the final overall output y[n]. The distributivity property asserts that this total output y[n] is identical to what you would obtain if you first determined an overall equivalent impulse response for the parallel connection, which is simply the sum of the individual impulse responses: hoverall [n]=h1 [n]+h2 [n]. Then, you convolve the original input x[n] with this hoverall [n] to get y[n].

Significance

This property greatly simplifies the analysis of parallel system structures and is fundamental for expressing complex system behavior as a sum of simpler, well-understood responses. It's also utilized in certain filter design techniques.

This property directly reflects the time-invariance characteristic of LTI systems. If y[n] is the output of convolving x[n] with h[n] (i.e., y[n]=x[n]∗h[n]), then:

• Shifting the input by n0 samples causes the output to be shifted by the same amount: x[n−n0 ]∗h[n]=y[n−n0 ]
• Similarly, shifting the impulse response by n0 samples also causes the output to be shifted by the same amount: x[n]∗h[n−n0 ]=y[n−n0 ]

Interpretation

This is a direct and intuitive consequence of time-invariance. If you delay the moment an input event occurs, the system's entire response pattern will simply be delayed by the exact same amount. The shape and amplitude of the output remain unchanged, only its position in time is shifted. This property underpins the ability of LTI systems to behave consistently regardless of when an input is applied.

Convolving any signal x[n] with a unit impulse located at n=0 leaves the signal unchanged.
x[n]∗δ[n]=x[n]

Interpretation

The unit impulse δ[n] acts as the "identity element" for the convolution operation. It behaves identically to how multiplying a number by 1 leaves the number unchanged. This makes sense: a system with an impulse response of δ[n] is essentially a "wire" or a "do-nothing" system; its output is always identical to its input.

Statement

Convolving any signal x[n] with a shifted unit impulse δ[n−n0 ] simply results in a shifted version of the original signal x[n].
x[n]∗δ[n−n0 ]=x[n−n0 ]

Interpretation

A system whose impulse response is a single shifted impulse (e.g., h[n]=δ[n−n0 ]) is a pure delay system. It simply delays its input by n0 samples. This property is fundamental and was directly used in the very derivation of the convolution sum itself, as it illustrates how each individual scaled impulse component within the input signal contributes a scaled and time-shifted version of the system's impulse response to form the total output.

For finite-duration sequences, this property helps predict the length of the output. If a finite-duration input signal x[n] is non-zero for Nx consecutive samples, and a finite-duration impulse response h[n] is non-zero for Nh consecutive samples, then the resulting output y[n]=x[n]∗h[n] will have a duration of (Nx +Nh −1) samples.

Example

Suppose x[n] is non-zero from n=0 to n=9 (so Nx =10 samples). And h[n] is non-zero from n=0 to n=4 (so Nh =5 samples). Then the output y[n] will have a duration of (10+5−1)=14 samples. If both signals start at index 0, the output will also start at index 0 and end at index 13.

Significance

This property is highly practical, as it provides a quick way to verify convolution results and, more importantly, helps in precisely determining the correct range of the output index n for which the convolution sum needs to be computed, thereby avoiding unnecessary calculations for zero-valued output samples.