Audio Signal Processing and Coding

2.2 SPECTRA OF ANALOG SIGNALS

The frequency spectrum of an analog signal is described in terms of the continuous Fourier transform (CFT). The CFT of a continuous-time signal, x(t), is given by

where ω is the frequency in radians per second (rad/s). Note that ω = 2πf, where f is the frequency in Hz. The complex-valued function, X(ω), describes the CFT magnitude and phase spectrum of the signal. The inverse CFT is given by

Figure 2.1. The pulse-sinc CFT pair.

Figure 2.2. CFT of a sinusoid and a truncated sinusoid.

The inverse CFT is also known as the synthesis formula because it describes the time-domain signal, x(t), in terms of complex sinusoids. In CFT theory, x(t) and X(ω) are called a transform pair, i.e.,

The pulse-sinc pair shown in Figure 2.1 is useful in explaining the effects of time-domain truncation on the spectra. For example, when a sinusoid is truncated then there is loss of resolution and spectral leakage as shown in Figure 2.2.

In real-life signal processing, all signals ...