Convolution

A mathematical technique called convolution can be used to combine two signals into a third signal. Convolution is therefore crucial to signals and systems since it links the input signal with the system’s impulse response to generate the output signal. To put it another way, an LTI system’s input-output relationship is expressed by convolution.

h(t) = T.[δ(t)]

for signal, x(t)

x(t)=∫^∞_−∞x(τ).δ(t−τ)dτ

Convolution Theorem

A system at rest (zero initial conditions) responds to any input by means of the convolution of that input and the system impulse response, according to the main convolution theorem.

let x₁(t), x₂(t) are the two signal then the convolution of the signal is defined as the

x(t) = x₁(t) ✻ x₂(t)

x(t) = ∫^∞_−∞ x₁(T) . x₂(t-T) dτ , −∞ < t < ∞

• Averaging : A moving average, which is a type of convolution, is frequently employed in time series analysis to reduce noise in data by substituting the average of nearby values in a moving window for a given data point. Since a moving average highlights a deeper underlying trend by eliminating short-term variations, it can be thought of as a low-pass filter.
• Smoothing : The process of smoothing aims to highlight long-term trends by removing short-term fluctuations from a signal. For instance, if you plotted a stock’s price changes every day, it would appear noisy; using a smoothing operator may make it clearer if the price was generally rising or falling over time.
• Basic Identity : It is easy to determine the first identity. Let f and g be two functions, and let α be a constant.

[αf ]✻g = f✻[αg] = α[ f✻g ]

Properties of Convolution

We have some basic property that help to reduce the complex calculation.

• Commutative Property: According to this property, the order in which two signals are convolved does not affect the outcome. x₁(t), x₂(t) is the input signal.

x₁(t) ✻ x₂(t) = x₂(t) ✻ x₁(t)

• Distributive Property: If there is a three signal x₁(t), x₂(t) and x₃(t) then distributive property satisfied the following condition.

x₁(t) ✻ [x₂(t) + x₃(t)] = [x₁(t) ✻ x₂(t)] + [x₁(t) ✻ x₃(t)]

• Associative Property: According to the convolution’s associative property, the arrangement of the signals within a convolution has no bearing on the outcome.

[f₁(t) ✻ f₂(t)] ✻ f₃(t) = f₁(t) ✻ [f₂(t) ✻ f₃(t)]

f₁(t), f₂(t) and f₃(t) are the signals

• Invertibility Property: if the x(t) is the signal then passes through the LTI system whose transfer function is h(t), then the output is passed through the another response whose transfer function is set to that level so we get x(t) as output.

so,

h(t) ✻ h1(t) = δ(t)

so the equivalent transfer function is δ(t) so we get x(t) -> x(t).

h1(t), h2(t) are the transfer function.

x(t) -> input signal.

Systems that are both linear and time-invariant are known as linear time-invariant systems, or LTI systems for short. When a system’s outputs for a linear combination of inputs match the outputs of a linear combination of each input response separately, the system is said to be linear. Time-invariant systems are ones whose output is independent of the timing of the input application. Long-term behavior in a system is predicted using LTI systems. The term “linear translation-invariant” can be used to describe these systems, giving it the broadest meaning possible. The analogous term in the case of generic discrete-time (i.e., sampled) systems is linear shift-invariant.