Usually when a signal is captured the domain is either temporal or spatial. For example, an audio signal from a microphone measures changes in air pressure (sound waves) over time, thus the measurements are in the **time domain**. An image sensor takes measurements from an array of photoreceptors, thus the signal exists in the **spatial domain**. A video signal is both in the time domain and the spatial domain. It can be helpful to deal with a signal in different domains, for example the **frequency domain**. Here the value of a signal at a given point represents the contribution to the signal from a *sinusoid* at a certain frequency. So instead of representing an amplitude at time , it represents the contribution to the signal by a sinusoid at frequency . The **Fourier Transform** transforms a signal into the frequency domain.

A **sinusoid** is any function that takes the form , where is the *amplitude*, is the *angular frequency*, and is the *phase*, as shown in Figure 1. The angular frequency is measured in radians per second, therefore is one full cycle, sometimes denoted for *frequency*, in cycles per second or hertz (Hz). One full cycle is called the *period*, .

Another way of representing a sinusoid is with a **complex exponential**, in the form . *Euler’s formula *tells us that this is equal to a complex number where the *real* part is and the *imaginary* part is . Just as we can represent a discrete-time signal as a linear combination (scaled, shifted) of unit impulses, a *periodic* discrete-time signal can be expressed as the linear combination of sinusoids. The **Discrete-Time Fourier Transform**, or **DTFT**, decomposes a discrete-time signal into the composite sinusoids. The sinusoids act as a *basis set* for the signal, in the same way that the and vectors form the basis set for 2-dimensional Cartesian space. The DTFT is given in the following equation:

In my previous article I mentioned that the discrete-time signal is derived from a continuous signal in the form , where is the *sampling period*. The inverse of the sampling period is the *sampling frequency*, which corresponds to the number of samples per second. Due to limitations of sampling a continuous signal, we often represent the frequencies in relation to the sampling frequency. This is called a **normalized frequency**, in terms of hertz it is the number of cycles per sample, equal to , or in terms of angular frequency it is the number of radians per sample, equal to . Since radians are a unit-less measurement the normalized angular frequency is sometimes referred to as inverse samples. For example, the following signal in Figure 2 can be decomposed into the sinusoids at the frequencies plotted in Figure 3; the sinusoids themselves are shown in Figure 4.

The **Fourier series** allows us to perform an **inverse Fourier Transform** to reconstruct a time-domain signal from the composite sinusoids using the complex exponential form:

Since the frequency domain representation is a sum of scaled complex exponentials, it is a complex function. The *magnitude* represents the amplitude of the sinusoid, while the *angle* or *argument* represents the phase.

Consider an LTI system with impulse response and output given inputs in the form of a complex exponentials:

In the last equation above, the function is called the **frequency response **of the system. We can see that is the Fourier transform of the impulse response . Also, instead of convolving the frequency response with the input, it is multiplied. In general we can relate the time-domain signals to their frequency domain representations as follows:

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