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


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 radians, which is 360
degrees (
). Therefore,
Hz is the same frequency as
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
).
For example, a periodic signal with a period of seconds has
a frequency of
Hz, and a radian frequency of
rad/s. The sampling rate,
, is the reciprocal of the
sampling period
, i.e.,



Sinusoids
The term sinusoid means a waveform of the type
Thus, a sinusoid may be defined as a cosine at amplitude






![$\displaystyle f(t) \isdef \frac{d}{dt} \theta(t) = \frac{d}{dt} \left[\omega t + \phi\right] = \omega
$](http://www.dsprelated.com/josimages_new/filters/img1302.png)




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 ) gives the
amplitude and phase of the sinusoidal signal-component at frequency
. Mathematically, the spectrum of a signal
is the Fourier
transform of its time-waveform. Equivalently, the spectrum is the
z transform evaluated on the unit circle
. 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 , its z transform
is called
and its spectrum is therefore
. The
time-waveform
is said to ``correspond'' to its z transform
,
meaning they are transform pairs. This correspondence is often denoted
, or
. 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 (or simply
) but
continuous frequency
(or
). 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 , meaning the whole time-waveform (
for all
), or
, to mean the entire spectrum taken as a whole.
Imagine, for example, that we have plotted
on a strip of paper
that is infinitely long. Then
refers to the complete
picture, while
refers to the
th sample point on the plot.
Next Section:
Complex and Trigonometric Identities
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Creating Minimum Phase Filters and Signals