Short-time Fourier transform of a guitar solo.
You don't have to be around signal processing very long before you realize that Fourier methods play a huge role in the field. Newcomers often wonder why they are so important. There are several very good reasons for the prominence of Fourier methods in signal processing. They offer substantial intuition, naturally follow from the way the physical world interacts with signals, and are amazingly useful for computation.
Fourier Methods in Signal Processing
There are multiple Fourier methods that are used in signal processing. The most common are the Fourier transform, the discrete-time Fourier transform, the discrete Fourier transform, and the short-time Fourier transform. Fourier methods are used for two primary purposes: mathematical analysis of problems and numerical analysis of data. The Fourier transform and discrete-time Fourier transform are mathematical analysis tools and cannot be evaluated exactly in a computer. The Fourier transform is used to analyze problems involving continuous-time signals or mixtures of continuous- and discrete-time signals. The discrete-time Fourier transform is used to analyze problems involving discrete-time signals or systems. In contrast, the discrete Fourier transform is the computational workhorse of signal processing. It is used solely for numerical analysis of data. Lastly, the short-time Fourier transform is a variation of the discrete Fourier transform that is used for numerical analysis of data whose frequency content changes with time.
You may be wondering why I haven't mentioned the fast Fourier transform algorithm or FFT. The FFT is not a distinct Fourier method, but is an efficient computational technique for evaluating the discrete Fourier transform.
Portrait of Joseph Fourier.
Fourier methods are named after Joseph Fourier, a French mathematician and physicist that lived from 1768 to 1830. He pioneered the use of sinusoids for representing arbitrary functions. An important problem of his day was understanding heat flow. Heat flow is governed by a partial differential equation called the heat equation. Fourier developed a technique for solving partial differential equations that assumed the solution was given by a weighted sum of harmonically related sinusoids. This work was controversial at the time and initially rejected by Lagrange, Laplace, and Legendre. Fourier persisted with his approach and his methods are now widely used in mathematics, science, and, of course, signal processing.
All Fourier methods use sinusoids of different frequencies as building blocks to represent arbitrary signals. The notion of frequency is widely understood. We often talk about high frequency and low frequency sounds. We are comfortable with the idea of different musical instruments having different frequency ranges, and with a complex musical piece being obtained by combining instruments. A musical score is very much like a Fourier decomposition in that it describes the music in terms of individual notes with different frequencies. These general concepts make it relatively easy to transition to the idea of representing an arbitrary signal as a sum of sinusoids of different frequencies.
Sunspot energy as a function of frequency.
Spectral analysis is an important aspect of signal processing. It is analogous to producing a score from a piece of music. The goal is to start with a signal and identify the strength of the sinusoidal components that make up the signal. The strength or amplitude of the sinusoids are displayed as a function of frequency. The importance of spectral analysis lies in the intuitive understanding it provides of the signal. For example, the figure at left depicts the amplitude of sinusoids as a function of frequency for measured sunspot activity from 1700 to 1987. We see a dominant component with frequency slightly less than 0.1 cycles per year, which suggests a periodic pattern of about 11 years in the activity.
Key Properties of Complex Sinusoids
The Fourier transform describes signals as weighted combinations of continuous-time complex-valued sinusoids, while the discrete-time Fourier transform, the discrete Fourier transform, and the short-time Fourier transform all employ discrete-time complex sinusoids. The vast utility of Fourier methods in signal processing - beyond intuition - is due to several powerful properties of complex sinusoids.
Differentiation of a continuous-time sinusoid results in a continuous-time sinusoid of the identical frequency. Only the amplitude and phase of the sinusoid are changed. Hence, differentiation of a sum of sinusoids of different frequencies results in a sum of sinusoids of the same frequencies. Only the amplitudes and phases of the sinusoids in the sum are changed. This property is important because of the significance of differentiation in the physical world. Electrical elements like capacitors and inductors relate voltages and currents through differentiation. Mechanical systems involving springs, masses, and dampers use differential equations to describe positions, velocities, and accelerations. Phenomena like heat, sound, electromagnetic waves, fluid flow, and elasticity are described with partial differential equations.
Time delay in computer-based signal processing plays an analogous role to differentiation in the physical world. Many signal-processing operations involve weighted combinations of delayed signals. It turns out that delay of a discrete-time complex sinusoid results in a discrete-time sinusoid of the identical frequency. Only the amplitude and phase are changed. Similarly, a delay of a sum of sinusoids of different frequencies results in a sum of sinusoids of the same frequencies. Only the amplitudes and phases of the sinusoids in the sum are changed.
Fourier Methods and Signal-System Interactions
Linear, time-invariant (LTI) systems are widely used to model effects in the physical world and are also widely used to manipulate signals in signal processing. A continuous-time LTI system is described as a weighted sum of derivatives of the signals. A discrete-time LTI system is described as a weighted sum of delayed signals. Hence, due to the differentiation and time delay properties, the output of an LTI system in response to a complex sinusoid input is a complex sinusoid of the same frequency. Only the amplitude and phase are changed. This is the so-called "eigenfunction" property. It follows that an input expressed as an arbitrary sum of complex sinusoids of different frequencies produces an output given by a sum of complex sinusoids of the same frequencies. The system only modifies the amplitude and phase of the individual sinusoids in the sum. The manner in which the system modifies the amplitudes and phase of the input sinusoids is called the frequency response of the system. This is an extremely powerful property and is introduced in the lesson Frequency Response Descriptions for LTI Systems.
Magnitude Response of a Low-Pass Filter.
Frequency response is a very intuitive description for the action of an LTI system on a signal. The magnitude of the frequency response tells us how the amplitudes of the input sinusoidal components are changed by the system. This gives rise to the idea of using systems to "filter" signals. A filter separates out certain frequency components of the input signal. The figure at left illustrates the magnitude of the frequency response (in dB) for a low-pass filter. Lower frequency sinusoids in the input signal see a gain of 0 dB or unity and are not attenuated. Higher frequency sinusoids experience gain less than -60 dB, that is, are multiplied by a factor of 10−3
or smaller, and are effectively eliminated from the input signal.
Fourier methods are natural tools for understanding and modeling the effects of the physical world on signals, and for designing and characterizing common signal processing systems.
The discrete Fourier transform may be computed very efficiently using an FFT algorithm. The computational efficiency of FFT algorithms is a direct consequence of the properties of complex sinusoids. The FFT algorithm was published by Cooley and Tukey in 1965. Computational power was quite limited in those early days of digital computing, and the FFT opened up possibilities for computer analysis of signals that were previously unimaginable. Thus, FFT algorithms fueled the rapid growth of the new field of signal processing. The FFT remains a cornerstone of modern signal processing as it allows solution of problems that are "large" with respect to available computing power.
Fourier methods are based on representing arbitrary signals as weighted sums of complex sinusoids. They are intuitive, apply to a large class of interesting signal processing systems and physical effects, and numerical Fourier analysis can be performed very efficiently. They play a huge role in signal processing and are worthy of study.