Introduction
What is signal processing? Sometimes it is like interpreting invisible hieroglyphs! A signal can be any information conveying meaning, such as sound, images or other data—even smoke signals. Here, signal processing means manipulating, analysing and interpreting signals in electrical or electronic systems. It involves mathematical algorithms and techniques for analysing and modifying signals, such as filtering, compression, enhancement and recognition.
Signal processing therefore plays a fundamental role in modern technologies and applications, including telecommunications, audio and video processing, biomedical engineering, control systems and computer engineering. Signals represent and communicate information; processing extracts and isolates useful information and/or improves signal quality.
Examples include removing background noise from a voice recording or smoothing an image with unwanted “noise”. It could also mean decoding an unclear input into a clear signal, as illustrated in Figure 1.
Figure 1: What appears to be random letters turns out to contain a hidden message after some signal processing (red letters).
*This illustration only demonstrates the concept
Figure 1 illustrates the concept by removing lowercase letters from the input to reveal a hidden message. In practice, this might mean removing unwanted frequencies from an input signal. This is how radios select frequencies and hence different channels. A radio receives electromagnetic radio waves transmitted by an antenna. Different frequencies carry information for different radio stations. Tuning a receiver to a particular frequency selects a signal and therefore a station. It is rather like decoding invisible hieroglyphs—sometimes that is signal processing!
In summary, signal processing aims to extract specific information from signals and convert it into the form required by the application.
This blog covers some basic transformations—mathematical algorithms and techniques—for signal processing. Before introducing them, let’s relate them to electronics.
Working with electronic signals often means working with time and frequency.
A time-domain representation shows signal values, such as voltage, over time. A frequency-domain representation describes the amplitudes and phases of frequency components. Amplitude and frequency are different properties: a particular voltage does not determine a particular frequency.
The mathematical algorithms and techniques introduced here are the Fourier transform, Laplace transform and Z-transform. They share similarities but also differ. Here is an overview of those similarities and differences.
Some words below are hyperlinks to sections of this blog that explain them further.
Domain
The Fourier transform has continuous-time and discrete-time forms; the DFT describes finite sampled sequences.
The Laplace transform is normally used for continuous-time signals; the Z-transform is the corresponding tool for discrete-time signals.
The Z-transform operates on discrete-time sequences.
Complexity and Mathematics
The Fourier transform uses complex exponential functions.
The Laplace transform uses an integral with a complex exponential kernel and a region of convergence. Complex logarithms are not a defining requirement.
The Z-transform uses a sum of sequence values weighted by powers of z and has a region of convergence.
Stability Analysis
Stability analysis is the part of systems and control theory used to study and predict whether a system is stable or unstable.
The Laplace transform is often used for stability analysis of linear time-invariant systems (LTI).
The Z-transform is used for stability analysis of digital filters (for example, a filter that removes background noise from a voice recording).
The Fourier transform describes frequency response where it exists; stability analysis usually also requires information about the impulse response or poles and region of convergence.
Frequency-Domain Representation
The Fourier transform maps a time signal to a frequency representation; continuous-time and discrete-time Fourier transforms have a continuous frequency variable, while the DFT uses discrete frequency bins.
The Laplace transform maps a time signal to the complex-frequency s-domain.
The Z-transform maps a discrete-time sequence to the complex z-plane. On the unit circle, where convergence allows, it gives the discrete-time frequency response.
Each transform has its own properties and applications, so the choice depends on the particular signal-processing problem.
What Are Continuous-Time and Discrete-Time Signals?
The difference concerns the time axis t in Figure 2. If the signal is defined for all values of time, t, it is continuous-time, as in Figures 2a and 2b.
For a discrete-time signal, values are defined only at discrete times rather than every value of t, as in Figures 2c and 2d.
Figure 2: Illustrates continuous-time and discrete-time signals, where
t = time
x(t) is a function of time
(a) A continuous-time sine wave, because t is defined at all times
(b) A two-level continuous-time waveform: its value is defined at every time, even though its amplitude has discrete levels.
(c) A sampled sine wave, defined only at discrete sample times.
(d) The sequence is defined only at discrete values of t, so (d) is a discrete-time signal.
What Is a Linear Time-Invariant System (LTI)?
Linearity means superposition: if inputs x1 and x2 produce outputs y1 and y2, then input a·x1+b·x2 produces a·y1+b·y2 for arbitrary constants a and b. This includes both addition and scaling.
Time invariance means that delaying an input by t0 delays the corresponding output by the same amount, without otherwise changing the response. Not all systems are linear or time-invariant.
Let’s illustrate superposition with signals. We input the red and blue waves separately into an LTI system and obtain the corresponding purple and turquoise outputs; see Figure 3.
Figure 3: Red and blue are inputs to an LTI system; purple and turquoise are their corresponding outputs.
If we then input the sum of the two previous inputs, we get the sum of the two previous outputs; see Figure 4.
Figure 4: The red and blue signals form one combined input; the purple and turquoise signals form its corresponding output.
This relationship between Figures 3 and 4 is the superposition principle, or superposition property. For a linear system, the response to a sum of stimuli equals the sum of the individual responses; scaling the stimuli scales the responses as well.
A nonlinear system does not satisfy superposition for all valid inputs and scalings. Its behaviour therefore cannot generally be reconstructed by simply adding individual responses.Let’s leave nonlinear systems aside for now. Taking one solid step at a time can be the fastest way forward.
What Is Complex Frequency?
Does complex frequency mean a frequency hidden in the fifth dimension, discovered by imagination? Unfortunately not! For the Laplace transform, complex frequency is s = σ + jω. The real part σ describes exponential growth or decay, and the imaginary part ω describes angular oscillation frequency. It does not simply mean “frequency with phase information”.Phase? What Is That?Even a single sinusoidal signal needs phase, as well as amplitude and frequency, to specify its position in time relative to a reference. Complex numbers conveniently represent sinusoidal amplitude and phase. This phasor representation is related to, but distinct from, the complex-frequency variable s. Complex arithmetic is a useful tool for simplifying valid calculations in electrical systems.
Complex numbers are commonly written a+ib, where a and b are real and i is the imaginary unit. They consist of a real and an imaginary part—or more precisely, one real number plus another real number multiplied by the imaginary unit i, with i² = -1. Electrical engineering often uses j instead of i to avoid confusion with current.
For example, for 2 + 3i, the real part Re is 2 and the imaginary part Im is 3. The imaginary term is 3i.
Fourier Transform
The Fourier transform decomposes signals into frequency components, representing them through sinusoidal functions or complex exponentials. This makes signals easier to analyse, manipulate and interpret.
It maps a signal from the time domain, where it is a function of time, to the frequency domain, where it is a function of frequency.
This helps isolate desired frequencies by filtering out those outside a chosen range.
Important Fourier representations include
the continuous-time Fourier transform (CTFT), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT).
The DTFT represents a discrete-time sequence using a continuous, periodic frequency variable. The DFT represents a finite sequence with a finite set of frequency bins.
The CTFT applies to continuous-time signals under the appropriate existence conditions.
Laplace Transform
The Laplace transform simplifies the mathematics of signal inputs and outputs. It transforms a continuous-time signal into the complex-frequency s-domain, where analysis and manipulation can be easier than solving the problem directly in time. The region of convergence is part of the transform description.
Z-Transform
The Z-transform is a mathematical tool for analysing and representing discrete-time signals. Like the Laplace transform, it can simplify the underlying mathematics. It maps a sequence from discrete time to a function of the complex variable z.
The Z-transform and its region of convergence support analysis and design of digital filters and other discrete-time systems. Its evaluation on the unit circle, when valid, connects it to frequency-domain analysis.
A key benefit is the ability to analyse and design digital filters used in telecommunications, audio and video processing and control systems. It also represents transfer functions of discrete-time LTI systems, making their behaviour easier to study.
Overall, Fourier, Laplace and Z-transforms are powerful, widely used mathematical tools that provide valuable insight into the behaviour and properties of many systems.
