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Signal Representation and Notation

Below is a summary of various notational conventions used in digital signal processing for representing signals and spectra. For a more detailed presentation, see the elementary introduction to signal representation, sinusoids, and exponentials in [84].A.1

Units

In this book, time $ t$ is always in physical units of seconds (s), while time $ n$ or $ m$ is in units of samples (counting numbers having no physical units). Time $ t$ is a continuous real variable, while discrete-time in samples is integer-valued. The physical time $ t$ corresponding to time $ n$ in samples is given by

$\displaystyle t = nT, $

where $ T$ is the sampling interval in seconds.

For frequencies, we have two physical units: (1) cycles per second and (2) radians per second. The name for cycles per second is Hertz (Hz) (though in the past it was cps). One cycle equals $ 2\pi$ radians, which is 360 degrees ($ \hbox{${}^{\circ}$}$). Therefore, $ f$ Hz is the same frequency as $ 2\pi f$ radians per second (rad/s). It is easy to confuse the two because both radians and cycles are pure numbers, so that both types of frequency are in physical units of inverse seconds (s $ \null^{-1}$).

For example, a periodic signal with a period of $ P$ seconds has a frequency of $ f = (1/P)$ Hz, and a radian frequency of $ \omega = 2\pi/P$ rad/s. The sampling rate, $ f_s$, is the reciprocal of the sampling period $ T$, i.e.,

$\displaystyle f_s = \frac{1}{T}. $

Since the sampling period $ T$ is in seconds, the sampling rate $ f_s=1/T$ is in Hz. It can be helpful, however, to think ``seconds per sample'' and ``samples per second,'' where ``samples'' is a dimensionless quantity (pure number) included for clarity. The amplitude of a signal may be in any arbitrary units such as volts, sound pressure (SPL), and so on.

Sinusoids

The term sinusoid means a waveform of the type

$\displaystyle A\cos(2\pi ft + \phi) = A \cos(\omega t + \phi). \protect$ (A.1)

Thus, a sinusoid may be defined as a cosine at amplitude $ A$, frequency $ f$, and phase $ \phi$. (See [84] for a fuller development and discussion.) A sinusoid's phase $ \phi$ is in radian units. We may call

$\displaystyle \theta(t) \isdef \omega t + \phi $

the instantaneous phase, as distinguished from the phase offset $ \phi$. Thus, the ``phase'' of a sinusoid typically refers to its phase offset. The instantaneous frequency of a sinusoid is defined as the derivative of the instantaneous phase with respect to time (see [84] for more):

$\displaystyle f(t) \isdef \frac{d}{dt} \theta(t) = \frac{d}{dt} \left[\omega t + \phi\right] = \omega $

A discrete-time sinusoid is simply obtained from a continuous-time sinusoid by replacing $ t$ by $ nT$ in Eq.$ \,$(A.1):

$\displaystyle A\cos(2\pi f nT + \phi) = A \cos(\omega n T + \phi). $

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 [84].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) [84], 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.

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