# Background Fundamentals

## 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

*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 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.*,

*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

Thus, a sinusoid may be defined as a

*cosine*at amplitude , frequency , and phase . (See [84] for a fuller development and discussion.) A sinusoid's

*phase*is in radian units. We may call

*instantaneous phase*, as distinguished from the

*phase offset*. 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):

*discrete-time sinusoid*is simply obtained from a continuous-time sinusoid by replacing by in Eq.(A.1):

### 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.

## Complex and Trigonometric Identities

This section gives a summary of some of the more useful mathematical identities for complex numbers and trigonometry in the context of digital filter analysis. For many more, see handbooks of mathematical functions such as Abramowitz and Stegun [2].

The symbol means ``is defined as''; stands for a complex number; and , , , and stand for real numbers. The quantity is used below to denote .

### Complex Numbers

### The Exponential Function

### Trigonometric Identities

#### Trigonometric Identities, Continued

### Half-Angle Tangent Identities

##
A Sum of Sinusoids at the

Same Frequency is Another

Sinusoid at that Frequency

It is an important and fundamental fact that a sum of sinusoids at the
same frequency, but different phase and amplitude, can always be
expressed as a *single* sinusoid at that frequency with some
resultant phase and amplitude. An important implication, for example,
is that

That is, if a sinusoid is input to an LTI system, the output will be a sinusoid at the same frequency, but possibly altered in amplitude and phase. This follows because the output of every LTI system can be expressed as a linear combination of delayed copies of the input signal. In this section, we derive this important result for the general case of sinusoids at the same frequency.

### Proof Using Trigonometry

We want to show it is always possible to solve

for and , given for . For each component sinusoid, we can write

(A.3) |

Applying this expansion to Eq.(A.2) yields

Equating coefficients gives

where and are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate , and (2) forming a ratio of both sides of Eq.(A.4) to eliminate . The results are

which has a unique solution for any values of and .

### Proof Using Complex Variables

To show by means of *phasor analysis* that Eq.(A.2) always has a solution, we can express each component sinusoid as

Thus, equality holds when we define

Since is just the polar representation of a complex number, there is always some value of and such that equals whatever complex number results on the right-hand side of Eq.(A.5).

As is often the case, we see that the use of Euler's identity and
complex analysis gives a simplified *algebraic* proof which
replaces a proof based on trigonometric identities.

### Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier

The heart of the preceding proof was the algebraic manipulation

*carrier term*``factors out'' of the sum. Inside the sum, each sinusoid is represented by a complex constant , known as the

*phasor*associated with that sinusoid.

For an arbitrary sinusoid having amplitude , phase , and radian frequency , we have

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Elementary Audio Digital Filters

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Conclusion