Spectrum

In this book, we think of filters primarily in terms of their effect on the spectrum of a signal. This is appropriate because the ear (to a first approximation) converts the time-waveform at the eardrum into a neurologically encoded spectrum. Intuitively, a spectrum (a complex function of frequency $\omega$) gives the amplitude and phase of the sinusoidal signal-component at frequency $\omega$. Mathematically, the spectrum of a signal $x$ is the Fourier transform of its time-waveform. Equivalently, the spectrum is the z transform evaluated on the unit circle $z=e^{j\omega T}$. A detailed introduction to spectrum analysis is given in [83].^A.2

We denote both the spectrum and the z transform of a signal by uppercase letters. For example, if the time-waveform is denoted $x(n)$, its z transform is called $X(z)$ and its spectrum is therefore $X(e^{j\omega T})$. The time-waveform $x(n)$ is said to ``correspond'' to its z transform $X(z)$, meaning they are transform pairs. This correspondence is often denoted $x(n)\leftrightarrow X(z)$, or $x(n)\leftrightarrow X(e^{j\omega T})$. Both the z transform and its special case, the (discrete-time) Fourier transform, are said to transform from the time domain to the frequency domain.

We deal most often with discrete time $nT$ (or simply $n$) but continuous frequency $f$ (or $\omega=2\pi f$). This is because the computer can represent only digital signals, and digital time-waveforms are discrete in time but may have energy at any frequency. On the other hand, if we were going to talk about FFTs (Fast Fourier Transforms--efficient implementations of the Discrete Fourier Transform, or DFT) [83], then we would have to discretize the frequency variable also in order to represent spectra inside the computer. In this book, however, we use spectra only for conceptual insights into the perceptual effects of digital filtering; therefore, we avoid discrete frequency for simplicity.

When we wish to consider an entire signal as a ``thing in itself,'' we write $x(\cdot)$, meaning the whole time-waveform ($x(n)$ for all $n$), or $X(\cdot)$, to mean the entire spectrum taken as a whole. Imagine, for example, that we have plotted $x(n)$ on a strip of paper that is infinitely long. Then $x(\cdot)$ refers to the complete picture, while $x(n)$ refers to the $n$th sample point on the plot.