Application of Convolution Properties Study Notes

By Neeraj Dubey|Updated : September 12th, 2022

In this article, you will find the Study Notes on Application of Convolution Properties which will cover the topics such as LTI systems, Causal and Anti-causal systems, Circular Convolution, Discrete Convolution, Circular Discrete Convolution, and application areas of Convolution.

Table of Content

Linear Time-Invariant Systems: A system can be regarded as the transformation of a signal according to some specified rule. Linear time-invariant systems are a special class of systems that can be described by constant-coefficient difference or differential equations or equivalently by a convolution sum or integral.

Classification of Systems: Systems can be classified by several properties. These include memory; invertibility; causality, stability; time invariance; and linearity.

Memory / Memoryless

A system without memory does not depend on past or future inputs. Instead, the output is determined entirely by the present input. An example is a resistor at low frequencies, where the voltage across its terminals is directly proportional to the current entering / leaving its terminals.

v(t)=Ri(t)

An example of a system with memory is a delay element:

y(t)=x(t-T)

Another example is a capacitor, which has a voltage across its terminals proportional to the time average of the current entering / leaving its terminals:

Invertibility

A system is invertible if there is a unique output for every input. In that way, the input signal can be inferred for any output signal. An example is a resistor with finite, non-zero resistance. The inverse system for the resistor is

An example of a non-invertible system is:

y(t)=tan[x(t)]

Causality

A causal system does not depend on future inputs. Thus a causal system cannot respond before the input is applied.

Examples of non-causal systems include image processing, where the independent variable is not time but space. In that case, the interpretation of future inputs refers to a point in space ahead of the present position. Another example is the post-processing of an audio signal. In that case, the signal is recorded onto some media before processing.

Stability

A system can be considered stable if, for any bounded input, the output is also bounded. An example of an unstable system is exponential population growth. Systems consisting of passive elements only, such as resistors and capacitors, are all stable systems. However, care must be taken when designing various control systems and filters to ensure they behave stably.

Time Invariance

A system is time-invariant if its characteristics do not change with time. As stated mathematically, if y(t) is the response to x(t), then the system is time-invariant if and only if for any given T, y(t-T) is the response to x(t-T) for all t.

Let

x(t) ⇔  y(t)

denote that y(t) is the system's response to x(t).

Suppose that

x1(t) ⇔  y1(t)

and

x2(t) ⇔  y2(t)

Then the system is linear if and only if

ax1(t)+bx2(t) ay1(t)+by2(t)

As an example, the system

y(t)=x(t)+1

is not linear.

Consider the signals x1(t) = 1 and x2(t) = 2. In that case, y1(t) = 2 and y2(t) = 3. The response to x(t) = x1(t)+x2(t) = 3 is y(t) = 4, which is clearly not equal to y1(t)+y2(t) = 5. Hence the system is non-linear.

There are very few truly linear systems in the physical world. However, many systems can be considered linear over a limited range of inputs and to a specified degree of accuracy or can be linearized through mathematical manipulation. An example is a thermocouple.

Causality: Not all linear time-invariant systems are necessarily causal. The impulse response reflects the causality of a system. For any causal system, the impulse response is zero for all t < 0.

Causality and Non-causality Properties

  • A causal system is a system in which the output depends only on current or past inputs but not future inputs.
  • An anti-causal system is a system in which the output depends only on current or future inputs but not past inputs. Finally, a non-causal system is a system in which the output depends on both past and future inputs.

The following figure shows impulse responses of discrete-time linear time-invariant causal, non-causal and anti-causal systems.

The impulse response of the causal, non-causal, and anti-causal system

  • Real-time systems in which time is the independent variable must always be causal because no system can depend on a future input value.
  • Non-causality can exist in domains such as image processing, where the independent variable is the pixel position.
  • A non-causal or anti-causal system can be converted to a causal system by introducing an appropriate delay.

Circular Convolution

When a function gT is periodic, with period T, then for functions, f, such that f∗gT exists, the convolution is also periodic and identical to

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where to is an arbitrary choice. Hence, the summation is called a periodic summation of the function f.

DISCRETE CONVOLUTION:

For complex-valued functions, f and g are defined on the set Z of integers, the discrete convolution of f and g is given by

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The convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers. When the sequences are the coefficients of two polynomials, then the coefficients of the ordinary product of the two polynomials are the convolution of the original two sequences. This is known as the Cauchy product of the coefficients of the sequences.

CIRCULAR DISCRETE CONVOLUTION:

When a function gN is periodic, with period N, then for functions, f, such that f∗gN exists, the convolution is also periodic and identical to

byjusexamprepThe summation on k is called a periodic summation of the function f. If gN is a periodic summation of another function, g, then
f∗gN is known as a circular convolution of f and g.

Application Areas of Convolution

Convolution and related operations are found in many applications in science, engineering, and mathematics:

  • In image processing: digital signal processing
    • In digital image processing, convolutional filtering plays an important role in many important algorithms in edge detection and related processes.
  • In digital data processing
    • Savitzky–Golay smoothing filters are used in analytical chemistry to analyze spectroscopic data. They can improve the signal-to-noise ratio with minimal distortion of the spectra. In statistics, a weighted moving average is a convolution.
  • In acoustics, reverberation is the convolution of the original sound with echoes from objects surrounding the sound source.

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