# Linear Time-Invariant Digital Filters

In this chapter, the important concepts of *linearity* and
*time-invariance* (LTI) are discussed. Only LTI
filters can be subjected to frequency-domain analysis as illustrated
in the preceding chapters. After studying this chapter, you should be
able to classify any filter as linear or nonlinear, and time-invariant
or time-varying.

The great majority of *audio* filters are LTI, for several
reasons: First, *no new spectral components* are introduced by
LTI filters. Time-*varying* filters, on the other hand, can
generate audible *sideband images* of the frequencies present in
the input signal (when the filter changes at audio rates).
Time-invariance is not overly restrictive, however, because the static
analysis holds very well for filters that change slowly with time.
(One rule of thumb is that the coefficients of a quasi-time-invariant
filter should be substantially constant over its impulse-response
duration.) *Nonlinear* filters generally create new sinusoidal
components at all sums and differences of the frequencies present in
the input signal.^{5.1}This includes both
*harmonic distortion* (when the input signal is periodic) and
*intermodulation distortion* (when at least two inharmonically
related tones are present). A truly linear filter does not cause
harmonic or intermodulation distortion.

All the examples of filters mentioned in Chapter 1 were LTI, or
approximately LTI. In addition, the transform and all forms of the
Fourier transform are linear operators, and these operators can be
viewed as *LTI filter banks*, or as a single LTI filter having
multiple outputs.

In the following sections, linearity and time-invariance will be formally introduced, together with some elementary mathematical aspects of signals.

## Definition of a Signal

Mathematically, we typically denote a signal as a real- or complex-valued function of an integer,

Definition.Areal discrete-time signalis defined as any time-ordered sequence of real numbers. Similarly, acomplex discrete-time signalis any time-ordered sequence of complex numbers.

*e.g.*, , . Thus, is the th real (or complex) number in the signal, and represents time as an integer

*sample number*.

Using the *set notation*
, and to denote
the set of all integers, real numbers, and complex numbers,
respectively, we can express that is a real, discrete-time signal
by expressing it as a function mapping every integer (optionally in
a restricted range) to a real number:

Similarly, a discrete-time *complex* signal is a mapping from
each integer to a complex number:

*i.e.*, ( is a complex number for every integer ).

It is useful to define as the *signal space* consisting
of all complex signals
,
.

We may expand these definitions slightly to include functions of the form , , where denotes the sampling interval in seconds. In this case, the time index has physical units of seconds, but it is isomorphic to the integers. For finite-duration signals, we may prepend and append zeros to extend its domain to all integers .

Mathematically, the set of all signals can be regarded a
*vector space*^{5.2} in
which every signal is a vector in the space (
). The
th sample of , , is regarded as the th *vector
coordinate*. Since signals as we have defined them are infinitely
long (being defined over all integers), the corresponding vector space
is *infinite-dimensional*. Every vector space comes with
a field of *scalars* which we may think of as *constant gain
factors* that can be applied to any signal in the space. For purposes
of this book, ``signal'' and ``vector'' mean the same thing, as do
``constant gain factor'' and ``scalar''. The signals and gain factors
(vectors and scalars) may be either real or complex, as applications
may require.

By definition, a vector space is *closed under linear
combinations*. That is, given any two vectors
and
, and any two scalars and , there exists a
vector
which satisfies
, *i.e.*,

A linear combination is what we might call a *mix* of two signals
and using mixing gains and (
). Thus, a *signal mix* is represented
mathematically as a *linear combination of vectors*. Since
signals in practice can overflow the available dynamic range,
resulting in *clipping* (or ``wrap-around''), it is not normally
true that the space of signals used in practice is closed under linear
combinations (mixing). However, in floating-point numerical
simulations, closure is true for most practical purposes.^{5.3}

## Definition of a Filter

Thus, a real digital filter maps every real, discrete-time signal to a real, discrete-time signal. A

Definition.Areal digital filteris defined as any real-valued function of a real signal for each integer .

*complex*filter, on the other hand, may produce a complex output signal even when its input signal is real.

We may express the input-output relation of a digital filter by the notation

where denotes the entire input signal, and is the output signal at time . (We will also refer to as simply .) The general filter is denoted by , which stands for any transformation from a signal to a sample value at time . The filter can also be called an

*operator*on the space of signals . The operator maps every signal to some new signal . (For simplicity, we take to be the space of complex signals whenever is complex.) If is linear, it can be called a

*linear operator*on . If, additionally, the signal space consists only of finite-length signals, all samples long,

*i.e.*, or , then every linear filter may be called a

*linear transformation*, which is representable by constant

*matrix*.

In this book, we are concerned primarily with *single-input,
single-output (SISO) digital filters*. For
this reason, the input and output signals of a digital filter are
defined as real or complex numbers for each time index (as opposed
to vectors). When both the input and output signals are
vector-valued, we have what is called a
*multi-input, multi-out (MIMO) digital filter*. We look at MIMO allpass filters in
§C.3 and MIMO state-space filter forms in Appendix G,
but we will not cover transfer-function analysis of MIMO filters using
*matrix fraction descriptions* [37].

## Examples of Digital Filters

While any mapping from signals to real numbers can be called a filter, we normally work with filters which have more structure than that. Some of the main structural features are illustrated in the following examples.

The filter analyzed in Chapter 1 was specified by

*difference equation*. This simple filter is a special case of an important class of filters called

*linear time-invariant (LTI) filters*. LTI filters are important in audio engineering because they are the

*only*filters that preserve signal frequencies.

The above example remains a real LTI filter if we scale the input
samples by any real *coefficients*:

If we use complex coefficients, the filter remains LTI, but it becomes
a *complex filter*:

The filter also remains LTI if we use more input samples in a shift-invariant way:

*non-causal*filter example. Causal filters may compute using only

*present and/or past input samples*, , , and so on.

Another class of causal LTI filters involves using *past output
samples* in addition to present and/or past input samples. The
past-output terms are called *feedback*,
and digital filters employing feedback are called
*recursive digital filters*:

An example *multi-input, multi-output* (MIMO)
digital filter is

The simplest *nonlinear* digital filter is

*i.e.*, it squares each sample of the input signal to produce the output signal. This example is also a

*memoryless nonlinearity*because the output at time is not dependent on past inputs or outputs. The nonlinear filter

Another nonlinear filter example is the
*median smoother* of order which assigns the middle value of
input samples centered about time to the output at time .
It is useful for ``outlier'' elimination. For example, it will reject
isolated noise spikes, and preserve steps.

An example of a linear *time-varying* filter is

These examples provide a kind of ``bottom up'' look at some of the
major types of digital filters. We will now take a ``top down''
approach and characterize *all* linear, time-invariant filters
mathematically. This characterization will enable us to specify
frequency-domain analysis tools that work for *any* LTI digital
filter.

## Linear Filters

In everyday terms, the fact that a filter is linear means simply that
the following two properties hold:

#### Scaling:

The amplitude of the output is proportional to the amplitude of the input (thescaling property).

#### Superposition:

When two signals are added together and fed to the filter, the filter output is the same as if one had put each signal through the filter separately and then added the outputs (thesuperposition property).

While the implications of linearity are far-reaching, the mathematical
definition is simple. Let us represent the general *linear*
(but possibly *time-varying*) filter as a *signal operator*:

where is the entire input signal, is the output at time , and is the filter expressed as a

*real-valued function of a signal*for each . Think of the subscript on as selecting the th output sample of the filter. In general,

*each*output sample can be a function of several or even

*all*input samples, and this is why we write as the filter input.

**Definition. **A filter
is said to be
*linear*
if for any pair of signals
and for all
constant gains , we have the following relation for each
sample time
:

where denotes the signal space (complex-valued sequences, in general). These two conditions are simply a mathematical restatement of the previous descriptive definition.

The
*scaling* property of linear systems
states that scaling
the input of a linear system (multiplying it by a constant gain
factor) scales the output by the same factor.
The
*superposition* property of linear systems states that
the response of a linear system to a sum of signals is the sum of the
responses to each individual input signal. Another view is that the
individual signals which have been summed at the input are processed
independently inside the filter--they superimpose and do not
interact. (The addition of two signals, sample by sample, is like
converting stereo to mono by mixing the two channels together
equally.)

Another example of a linear signal medium is the earth's
atmosphere. When two sounds are in the air at once, the air pressure
fluctuations that convey them simply add (unless they are extremely
loud). Since any finite continuous signal can be represented as a sum
(*i.e.*, superposition) of sinusoids, we can predict the filter response to
any input signal just by knowing the response for all
sinusoids. Without superposition, we have no such general description
and it may be impossible to do any better than to catalog the filter
output for each possible input.

Linear operators distribute over linear combinations, *i.e.*,

#### Real Linear Filtering of Complex Signals

When a filter is a linear filter (but not necessarily time-invariant), and its input is a complex signal , then, by linearity,

Appendix H presents a linear-algebraic view of linear filters that can be useful in certain applications.

## Time-Invariant Filters

In plain terms, a *time-invariant
filter* (or *shift-invariant
filter*) is one which performs the
*same operation at all times*. It is awkward to express this
mathematically by restrictions on Eq.(4.2) because of the use of
as the symbol for the filter input. What we want to say is
that if the input signal is delayed (shifted) by, say, samples,
then the output waveform is simply delayed by samples and
unchanged otherwise. Thus , the output waveform from a
time-invariant filter, merely *shifts* forward or backward in
time as the input waveform is shifted forward or backward
in time.

**Definition. **A digital filter
is said to be
*time-invariant*
if, for every input signal , we have

where the -sample

*shift operator*is defined by

## Showing Linearity and Time Invariance, or Not

The filter is nonlinear and time invariant. The scaling property of linearity clearly fails since, scaling by gives the output signal , while . The filter is time invariant, however, because delaying by samples gives which is the same as .

The filter
is linear and *time varying*.
We can show linearity by setting the input to a linear combination of
two signals
, where and
are constants:

Thus, scaling and superposition are verified. The filter is time-varying, however, since the time-shifted output is which is not the same as the filter applied to a time-shifted input ( ). Note that in applying the time-invariance test, we time-shift the input signal only, not the coefficients.

The filter , where is any constant, is *nonlinear*
and time-invariant, in general. The condition for time invariance is
satisfied (in a degenerate way) because a constant signal equals all
shifts of itself. The constant filter *is* technically linear,
however, for , since
, even though the input
signal has no effect on the output signal at all.

Any filter of the form
is linear and
time-invariant. This is a special case of a *sliding linear
combination* (also called a *running weighted sum*, or
*moving average* when
).
All sliding linear combinations are linear,
and they are time-invariant as well when the coefficients (
) are constant with respect to time.

Sliding linear combinations may also include past *output*
samples as well (feedback terms). A simple example is any filter of
the form

Since linear combinations of linear combinations are linear combinations, we can use

*induction*to show linearity and time invariance of a constant sliding linear combination including feedback terms. In the case of this example, we have, for an input signal starting at time zero,

If the input signal is now replaced by , which is delayed by samples, then the output is for , followed by

or for all and . This establishes that each output sample from the filter of Eq.(4.7) can be expressed as a time-invariant linear combination of present and past samples.

##
Nonlinear Filter Example:

Dynamic Range Compression

A simple practical example of a
*nonlinear* filtering operation
is *dynamic range compression*, such as occurs in Dolby or DBX
noise reduction when recording to magnetic tape (which, believe it or
not, still happens once in a while). The purpose of dynamic range
compression is to map the natural dynamic range of a signal to a
smaller range. For example, audio signals can easily span a range of
100 dB or more, while magnetic tape has a linear range on the order of
only 55 dB. It is therefore important to compress the dynamic range
when making analog recordings to magnetic tape. Compressing the
dynamic range of a signal for recording and then expanding it on
playback may be called
*companding*
(compression/expansion).

Recording engineers often compress the dynamic range of individual tracks to intentionally ``flatten'' their audio dynamic range for greater musical uniformity. Compression is also often applied to a final mix.

Another type of dynamic-range compressor is called a *limiter*,
which is used in recording studios to ``soft limit'' a signal when it
begins to exceed the available dynamic range. A limiter may be
implemented as a very high compression ratio above some amplitude
threshold. This replaces ``hard clipping'' by ``soft limiting,''
which sounds less harsh and may even go unnoticed if there were no
indicator.

The preceding examples can be modeled as a variable *gain* that
automatically ``turns up the volume'' (increases the gain) when the
signal level is low, and turns it down when the level is high. The
signal level is normally measured over a short time interval that
includes at least one period of the lowest frequency allowed, and
typically several periods of any pitched signal present. The gain
normally reacts faster to attacks than to decays in audio compressors.

### Why Dynamic Range Compression is Nonlinear

We can model dynamic range compression as a *level-dependent
gain*. Multiplying a signal by a constant gain (``volume control''),
on the other hand, is a linear operation. Let's check that the
scaling and superposition properties of linear systems are satisfied
by a constant gain: For any signals , and for any constants
, we must have

Dynamic range compression can also be seen as a *time-varying
gain* factor, so one might be tempted to classify it as a linear,
time-varying filter. However, this would be incorrect because the
gain , which multiplies the input, *depends on the input
signal* . This happens because the compressor must estimate the
current signal level in order to normalize it. Dynamic range
compression can be expressed symbolically as a filter of the form

*rms level*(the ``root mean square'' [84, p. 75] computed over a sliding time-window). Since many successive samples of are needed to estimate the current level, we cannot correctly write for the gain function, although we could write something like (borrowing matlab syntax), where is the number of past samples needed to estimate the current amplitude level. In general,

In general, any signal operation that includes a multiplication in which both multiplicands depend on the input signal can be shown to be nonlinear.

## A Musical Time-Varying Filter Example

Note, however, that a gain may vary with time *independently*
of to yield a linear *time-varying* filter. In this case,
linearity may be demonstrated by verifying

*tremolo*function, which can be written as a time-varying gain, . For example, would give a maximally deep tremolo with 4 swells per second.

## Analysis of Nonlinear Filters

There is no general theory of nonlinear systems. A nonlinear system with memory can be quite surprising. In particular, it can emit any output signal in response to any input signal. For example, it could replace all music by Beethoven with something by Mozart, etc. That said, many subclasses of nonlinear filters can be successfully analyzed:

- A nonlinear, memoryless, time-invariant ``black box'' can be ``mapped out'' by measuring its response to a scaled impulse at each amplitude , where denotes the impulse signal ( ).
- A memoryless nonlinearity followed by an LTI filter can similarly be
characterized by a stack of impulse-responses indexed by amplitude (look up
*dynamic convolution*on the Web).

One often-used tool for nonlinear systems analysis is Volterra series [4]. A Volterra series expansion represents a nonlinear system as a sum of iterated convolutions:

*Volterra kernels*. The special notation indicates that the second-order kernel is fundamentally two-dimensional, meaning that the third term above (the first nonlinear term) is written out explicitly as

In the special case for which the Volterra expansion reduces to

## Conclusions

This chapter has discussed the concepts of linearity and
time-invariance in some detail, with various examples considered. In
the rest of this book, all filters discussed will be linear and (at
least approximately) time-invariant. For brevity, these will be
referred to as *LTI filters*.

## Linearity and Time-Invariance Problems

See `http://ccrma.stanford.edu/~jos/filtersp/Linearity_Time_Invariance_Problems.html`

**Next Section:**

Time Domain Digital Filter Representations

**Previous Section:**

Analysis of a Digital Comb Filter