Analog and Digital Technologies
Linear discrete time-invariant systems

Linear time-invariant system (LTI) has direct applications in seismology, circuits, signal processing, control theory, and other technical areas. The analysis of the continuous-time LTI and discrete-time LTI are rather similar, but the discrete-time case involves fewer technicalities, so we concentrate on it.

LDTI (Linear discrete time-invariant system) takes one discrete input signal and produce one discrete output signal with the following properties:

- The system is linear. It means that if the input signals x1(n) and x2(n) generate output signals y1(n) and y2(n) and if a1 and a2 are constants, then the input signal a1x1(n)+ a2x2(n) generate output signal a1y1(n)+ a2y2(n).
- The system is time invariant. This means that if the input signal x(n) generates the output signal y(n), then, for each real number s, the time shifted input signal generates the time shifted output .

## The difference equation

The first important fact concerning the behavior of linear discrete time-invariant system is that the response of the system to any input is completely determined by its response to one special input, the Kronecker delta impulse (defined in chapter Important signals) at time 0. Let’s denote the output by h(n) that results from the Kronecker delta impulse at time 0. It is called the impulse response of the system. We’ll now define a formula that expresses the output generated by any input x(n).

This equation describes recursive LDTI. Samples of output signal are linear combination of input signal and weighing coefficients ak and bk. Systems described by this equation are with infinite impulse response (IIR).

Special case is differential equation for non recursive system. In this case the output signal is dependent only on input signal, not on previous samples of output signal. This system is with finite impulse response (FIR) and mathematically is described as:

## Convolution

Convolution is other possibility how to describe LDTI. If the impulse response h(n) of LDTI is known and input signal is x(n), than the output signal is given as:

where operator * is convolution product. The length of the output signal is defined as where is the length of the input signal and is length of the impulse response.

The principle is based on superposition in linear systems. The output signal is given as sum of weighted and shifted impulse responses. For easy understanding please read following example.

Let’s have FIR system with impulse response h(n) = {1, 2, 3}. Input signal is given as x(n) = {x(0), x(1), x(2), x(3)}.

Output signal is calculated by using convolution.

The length of h(n) is Dh =3 and length of input signal x(n) is Dx =4, then length of output signal is Dy =6. Convolution product can by calculated in table.

Convolution
n | 0 | 1 | 2 | 3 | 4 | 5 | 6
x(0) | x(0) | 2 x(0) | 3 x(0) | 0 | 0 | 0 | 0
x(1) |  | x(1) | 2 x(1) | 3 x(1) | 0 | 0 | 0
x(2) |  |  | x(2) | 2 x(2) | 3 x(2) | 0 | 0
x(3) |  |  |  | x(3) | 2 x(3) | 3x(3) | 0
y(n) | y(0) | y(1) | y(2) | y(3) | y(4) | y(5) | 0

In each table row are samples of input signal weighted with samples of impulse response. Rows are shifted right so it corresponds to delay of input signal. In last row are samples of output signal given as superposition of values in columns for n = 0, 1, 2,... For example :

y(1) = 2.x(0) + x(1)

y(2) = 3.x(0) + 2.x(1) + x(2), etc.

To get convolution of two signals, one signal is sort in reverse order and the other one is shifted from left to right. For each step the sum of products is calculated.

## Transfer function

Transfer function represents the relation between the input and output of a LDTI system with zero initial conditions in frequency domain. Transfer function can be derivate from difference equation or impulse response. In both cases Discrete Fourier transformation is used. It was defined before . After DFT of each component we get

Relation between input and output signal in time and frequency domain

It’s obvious that h(n) and characterize the same system in different domains. In case DFT is applied to difference equation, transfer function is described by using weighted coefficient ak and bk which are the same as in difference equation:

.

Formula in fraction form is more convenient because the result of dividing numerator by denominator is samples of impulse response. In case of IIR system the number of samples is infinite. For FIR systems the formula is not fraction, because denominator is equal 1.

The transfer function is very important and based on frequency response is estimated. This is used mainly in filter theory.

### Frequency response

Frequency response is used to characterize the dynamics of the system.

It is a measure of amplitude (or magnitude) and phase of the output as a function of frequency, in comparison to the input.

In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input.

So for frequency response of transfer function is defined:

where and .

The absolute value of transfer function is called magnitude frequency characteristic and is phase frequency characteristic. is not continuous function but function with step changes about 180 degrees. These steps can be removed by change of mark of magnitude frequency characteristic “–“ from “+” to – and vice versa. This change is done for each step change in . New phase frequency characteristic Ɵ(Ω) is continuous function. The relation between amplitude and magnitude frequency characteristics following:

.

Based on information above following is defined for frequency characteristics:

Examples of transfer function of LDTI, magnitude and phase frequency characteristic (one and more periods)