Signals and Systems

System Properties

In this section, several properties of systems are described based on their mathematical characteristics.

Linear and nonlinear

Simply put, a system is linear if superposition holds. Superposition refers to the ability of a system to process signals individually and then sum them up to process all the signals simultaneously.

In practical terms, think about a karaoke system. You want the audio amplifier that drives the speakers in this kind of system to be linear so the music and singer’s voice in the microphone can merge without causing distortion, which happens with a nonlinear amplifier. On the other hand, many hard rock guitar players send their signal through a nonlinear amplifier to get some distortion.

Time-invariant and time varying

A system is time-invariant if its properties or characteristics don’t change with time. A mathematical statement of this is that given y(t) = T{x(t)}, and any time offset t₀, the time-shifted input x₁(t) = x(t - t₀) must produce system output y₁(t) = y(t - t₀)

Here, y(t) is the system output to the present input x(t). For time invariance to hold, the output of a system is unchanged (except for the time offset by t₀) when you apply the same input at any arbitrary offset t₀.

A system that doesn’t obey the condition established for time invariance is said to be time varying. Creating a system with a time-varying property is as easy as twisting the volume control on your car stereo. Specifically, the gain of the system is time varying.

A noise-removing filter is typically designed to be time-invariant. Assuming the noise signal characteristics and the desired signal characteristics are fixed, the filter design should be time-invariant. A time-varying filter, known as an adaptive filter, is needed when the noise signal characteristics change over time. Think of noise-canceling headphones that give you relative peace and quiet riding in an airplane or on the flight deck of an aircraft carrier. These headphones are a time-varying system.

Causal and non-causal

A system that is causal is nonanticipative; that is, the system can’t anticipate the arrival of a signal at the input. Sounds crazy, I know, but a non-causal system can predict the future (in a signals sense); it anticipates the signal input. Mathematically, you can define such a system, but building a physical system is impossible.

A system is causal if all output values, y(t₀), depend only on input values x(t) for t less than or equal t₀ — or the present output depends only on past and present input values. A non-causal system is more of a mathematical concept than a practical reality. A system that can use future values of the input to form the present output can predict the future.

With discrete-time signals and systems, it’s possible to store a signal in memory and then process it later by using a non-causal system. The catch is that the processing is all being done by using past values of the input, because you’re working with a recorded signal. The math of a non-causal system is still at work, because the system doesn’t realize that the signal was prerecorded. For continuous-time systems, making this work is harder; you can perform non-causal processing on continuous-time signals with records and tapes of music recordings.

Memory and memoryless

Very simply, a system is memoryless if each output y(t) depends only on the present input x(t). Can a memoryless system be non-causal? If the output depends only on the present input, then no way can the future values of the input be used to form the present output. Yet causal systems aren’t necessarily memoryless. A causal system can utilize past values of the input in forming the present output.

A system that filters a system generally does so by using the present and past values of the input to form the present output. A system described by a linear constant coefficient (LCC) differential equation is one such example. An electronic circuit that’s composed of resistors, capacitors, and inductors is another example. The capacitors and inductors are the memory elements. A system with only resistors has no memory.

Bounded-input bounded-output

A system is bounded-input bounded-output (BIBO) stable only if every bounded input produces a bounded output. What’s this bounded stuff? Bounded is a mathematical term that means a signal has magnitude less than infinity over all time. The signal x(t) — which may be an input or an output — is bounded if some positive constant B_x exists such that the |x(t)| is less or equal B_x.

To show that the property holds for any bounded input is the fundamental challenge of this scenario. [The qualifier] any represents quite a lot of cases; testing them all can be prohibitive. Therefore, some proof-writing skills are required here!

Choosing Linear and Time-Invariant Systems

From a design and analysis standpoint, engineers are typically most interested in working with systems that are both linear and time-invariant because such systems can meet demanding real-world requirements and allow for smoother analysis in the time, frequency, and s-domains. The ability to analyze system performance is critical; you want to be confident that your design meets requirements before committing to expensive prototypes.

Convolution

(Heavily from https://www.mathworks.com/discovery/convolution.html)

Convolution is a mathematical operation that combines two functions to describe the overlap between them. Convolution takes two functions and slides one of them over the other, multiplying the function values at each point where they overlap, and adding up the products to create a new function. This process creates a new function that represents how the two original functions interact with each other.

Formally, convolution is an integral that expresses the amount of overlap of one function, f(t), as it is shifted over function g(t), expressed as:

(f * g) (t) \approx^{d e f} \int_{- \infty}^{\infty} f (τ) g (t - τ) d τ

Depending on the application of convolution, functions may be replaced with signals, images, or other types of data.

Convolution in Signal Processing

Convolution is used in digital signal processing to study and design linear time-invariant (LTI) systems such as digital filters.

The output signal, $y [n]$ , in LTI systems is the convolution of the input signal, $x [n]$ and impulse response $h [n]$ of the system.

In practice, the convolution theorem is used to design filters in the frequency domain. The convolution theorem states that:

Convolution in the time domain is equivalent to multiplication in the frequency domain.

The frequency domain can also be used to improve the execution time of convolutions. Using the FFT algorithm, signals can be transformed to the frequency domain, multiplied, and transformed back to the time domain. For signals of large enough size, this can result in a substantial speed increase.