1 Introduction to the linear time-invariant systems

Linear filters, despite their nature (mechanical, electronic, optic etc.), can be considered as linear dynamical systems, the behaviour of which can be described by ordinary differential equations.The input and output signals of the filter with lamped parameters are connected by an equation

A transfer function of a dynamical system is defined as a Laplace transform of the output signal divided by the Laplace transform of the input signal. Transfer functions provide the mathematical representation of linear dynamical systems with lamped parameters.

For the valid linear time-invariant systems and are rational polynomials in .

Given the complex functions and of real argument and their Laplace transforms and respectively:

Some properties of a Laplace transform:

  • Linearity

  • Shifting in t-domain

  • Scaling in t-domain

  • Shifting in s-domain

  • Differentiation in time domain

  • Differentiation in s-domain

  • Integration in time domain

  • Intergation in s-domain

  • Convolution

Examples of Laplace transformations:


Block diagram representations

Filters, as well as control systems, can be portrayed in the form of block diagrams. A block diagram is the conventional picture in which individual elements of the system are presented by rectangles and links between elements.

A common methodology for the representation of dynamic systems (control systems, filters, etc) introduces the model element that is usually described by the differential equations of the first or second order. On the block diagram, all blocks are depicted as rectangles, with a transfer function inside of it (Fig. 1.1).

Fig. 1.1 System has a frequency domain transfer function

Fig. 1.2 System has a time domain impulse response

A complex structural system can be depicted using three basic types of connections: parallel connection, serial connection, and network with feedback.


Parallel connection

In a parallel connection, the input signals of all blocks are identical and equal to the input . The output signal is equal to the sum of the output signals of all blocks. The parallel system, which has a frequency domain transfer function, can be portrayed as follows:

Fig. 1.3 Parallel connection of a system with frequency domain transfer functions.

The transfer function of block systems connected in parallel (see Fig. 1.3) can be obtained as follows:

Consequently, if the network consists of blocks connected in parallel, the equivalent transfer function is computed as follows:

The time domain parallel system with an impulse response can be portrayed as follows :

Fig. 1.4 Parallel connection of a system with time domain transfer functions

The transfer function of such a system can be obtained as follows

Consequently, if the network consists of blocks connected in parallel, the equivalent impulse response is computed as follows


Serial connection

In a serial connection, the output of each block (except the last) is connected to the input of the next one. Serial systems with a frequency domain transfer function can be portrayed as follows:

Fig. 1.5 Serial connection of a system with frequency domain transfer functions

The equivalent transfer function of a serial connection (Fig. 1.5) can be obtained as follows

Consequently, if blocks are connected in series, the equivalent transfer function is computed as follows :

The same serial system with a time domain impulse response can be portrayed as follows :

Fig. 1.6 Serial connection of a system with time domain transfer functions

The equivalent transfer function is :

So if blocks are connected in series, the equivalent impulse response is computed as follows :


Network with feedback

A network where an output signal is algebraically summed with an input signal is called a network with feedback.

Fig. 1.7 Network with feedback

The following equations describe the network ( Fig.1.7)

Solving the last equation for yields:

Consequently, the equivalent transfer function of a network with feedback can be computed as follows:

Parameter in (1.20) characterizes the type of feedback in the network.


Transfer functions properties

The transfer function obtained for a stable time-invariant linear filter with lamped parameters has to have certain properties:

  • All coefficients in and are real positive numbers

  • The order of the filter is the highest power of in denominator

  • The highest power of in nominator is less or equal to the filter's order

  • Zeros of denominator - are the poles of a transfer function. Poles can be real negative numbers, or complex conjugate pairs with negative real parts. This is necessary for the filter to be stable.

Therefore, the transfer function of a linear filter can be written in the form

where the zeros and poles are either real numbers or complex conjugate pairs.

Let us consider the functions:

Evaluating these functions at yields:

Using these properties, it can be easily shown that

The following properties can be obtained from (1.26):

The regular form of a transfer function of a time-invariant linear filter can be represented as a ratio of polynomials in .

The general factored form of a transfer function represented by zeros and poles can be written in the form (1.23). Since all complex roots (zeros and poles) occur in complex conjugate pairs, the transfer function can be represented as , where represents the factors with real roots (zeros and poles), and represents the factors with complex conjugate roots.

Both the nominator and denominator of a transfer function can be represented as linear and quadratic form factors with real coefficients.

Factored form (1.31) of the transfer function is commonly used to represent linear filters with lamped parameters. The order of the filter is a highest power of in denominator and is equal to .


Magnitude response

The magnitude response (or the gain) of a filter is an absolute value of the transfer function evaluated at .

The magnitude response can also be represented in the form (1.26)

The magnitude response is simply a ratio of an output signal to an input signal. This ratio is very small in the stopband and close to a unit in the passband.

It is a common practice to measure the magnitude response in a logarithmic scale:

If the logarithm in (1.33) is negative and, consequently, the gain is also negative.

The gain is defined as a ratio of an output signal to an input signal. The attenuation of a signal is a ratio of an input signal to an output. In the logarithmic scale, attenuation is connected to the gain as follows:

In the logarithmic scale, attenuations and gains are measured in decibels.


Phase response

The transmission of a signal through a filter is not completely described by attenuation. The phase characteristics are very important for obtaining a precise picture of signal transitions. Phase characteristics of a filter establish the relationship between the phase of input and output signals. Phase response is typically a shift of the phase between output and input signals.

The transfer function of a filter represented in a polar coordinate system can be determined as follows:

Phase response can be found from the expression


Phase delay

The sinusoid input to any linear system results in sinusoid output. If the input signal is

then the output signal can be expressed as follows

In this expression is the phase response of the system. Expression (1.37) can also be written in the form

Parameter in (1.38) is the phase delay. The phase delay closely corresponds to the phase response. If the phase response characterizes the phase shift between output and input signals, the phase delay characterizes the time delay between the output and input signals.


Group delay

The phase delay characterizes the time delay between the input and output signals of the system at a given frequency. Usually the signals have a wide frequency range, and the phase delays of the signals through a filter are different for the different frequencies. The input signal to the filter can be presented as

For the slowly changing amplitude of the input signal, the output signal can be expressed as follows:

In this expression parameter

is a group delay, and is a phase delay.

So, by definition the group delay is a derivative of phase.

It is desirable for the systems to have a flat group delay, when delays at all frequencies are the same. Mathematically it can be realized if the transfer function of the filter has the linear phase response

In this case


Impulse response

The analog delta function (or Dirac delta function) is defined as follows

Impulse response of a dynamic system is defined as an output of the system when the input signal has the properties (1.41). Impulse response completely describes the behavior of the linear dynamic system in the time domain. If the input signal of the system is given by the function , then the output signal can be expressed as a convolution of and

The transfer function of a linear system is defined as a ratio of the Laplace transform of an output signal to the Laplace transform of an input signal. Also, it can be shown that a transfer function of a system can be expressed as a Laplace transform of an impulse response.

Consequently, if the impulse response of a system is known, the transfer function can be found by making a Laplace transform of the impulse response. If the transfer function of a system is known, the impulse response can be found as an inverse Laplace transform of the transfer function.

For the transfer function that can be expressed as a ratio of polynomials

and the highest power of in nominator is less than the highest power of in denominator , the inverse Laplace transform is given by Heaviside expansion

where poles of , - shows how many times the root is repeated(see [1], [2]). Expression (1.45) can be significanly simplified if the transfer function doesn't have repeated roots

Note, that in (1.46) degree of polynomial must be less than a degree of polynomial . Important application of (1.46) related to the case when the transfer function can be expressed as

where the degree of polynomial is not higher(less or equal) than the degree of polynomial . In this case, expression (1.46) can be rearranged to the form

Note that the summation in this expression is made for all the roots of .

Another simplification of (1.46) can be done for the case when all coefficients in and are real numbers. Expression (1.46) can be rearranged to the following form

where the first summation is for real poles, and the second summation is for complex poles with positive imaginary parts.

Expression (1.46) and simplifications (1.48),(1.49) can be used for the determination of an impulse and step response when the transfer function is given as a ratio of polynomials.


Step response

Unit step function (or Heaviside step function) is defined as a function, which has a zero value for negative argument and a unit value for positive argument.

Fig. 1.8 Unit step function.

Unit step function can be represented as an integral of the Dirac delta function

Step response of a dynamic system is defined as an output of the system when the input signal is a unit step function. Step response is closely related to the impulse response and it can be expressed as follows

The connection between a step response and a transfer function of the system can be found from the integration properties of a Laplace transform (see [1], [2])

So, if is a transfer function of a system, the connections between a step response and a Laplace transform of the step response are expressed as follows

and

As it was shown above, the transfer function of a linear filter can be represented as a ratio of polynomials , so

An impulse response can be expressed as a derivative of a step response


References

[1] M. A. Lavrentyev, B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1973 (Russian)

[2] A. Markushevich, Theory of Functions of a Complex Variable, second edition, Chelsea Publishing Company, 1965


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