Linear Systems Models

Transfer Function

The transfer function is the basic z-domain representation of a a digital filter, expressing the filter as a ratio of two polynomials. The transfer function model description for the z-transform of a difference equation is

The constants b(i) and a(i) are the filter coefficients, stored in two row vectors.

Example

A simple IIR filter is

can be defined in Matlab by

b = [2 3 4];
a = [ 1 3 3 1];

This example will be used to illustrate other representations of linear systems.

Zero-Pole-Gain

The factored or zero-pole-gain form of a transfer function is

By convention, Matlab stores polynomial coefficients in row vectors and polynomial roots in column vectors. In zero-pole-gain form, therefore, the zero and pole locations for the denominator and numerator of a transfer function reside in column vectors. The factored transfer function gain k is a scalar.

The poly and roots functions convert between polynomial and zero-pole-gain representations

Example

The zeros and poles of the example filter are

q = roots(b)

q =

  -0.7500 + 1.1990i
  -0.7500 - 1.1990i

p = roots(a)

p =

  -1.0000          
  -1.0000 + 0.0000i
  -1.0000 - 0.0000i

k = b(1)/a(1)

k =

     2

The Matlab signal-processing function tf2zp also does transfer function to zero-pole-gain form.

[z, p, k] = tf2zp(b,a)

z =

  -0.7500 + 1.1990i
  -0.7500 - 1.1990i


p =

  -1.0000          
  -1.0000 + 0.0000i
  -1.0000 - 0.0000i


k =

     2

Returning to the original polynomials:

bb = k*poly(q)

bb =

    2.0000    3.0000    4.0000

aa = poly(p)

aa =

    1.0000    3.0000    3.0000    1.0000

Note that b and a in this case represent the transfer function

For b = [2 3 4], the roots function misses the zero for z equal to 0. In fact, it misses poles and zeros for z equal to 0 whenever the input transfer function has more poles than zeros or vice versa. This is acceptable in most cases. To circumvent the problem, however, simply append zeros to make the vectors the same length before using the roots function, for example, b = [b 0].

[z,p,k] = tf2zp([b 0],a)

z =

        0          
  -0.7500 + 1.1990i
  -0.7500 - 1.1990i


p =

  -1.0000          
  -1.0000 + 0.0000i
  -1.0000 - 0.0000i


k =

     2

State-Space

It is always possible to represent a digital filter, or a system of difference equations, as a set of first-order difference equations. In matrix or state-space form, you can write the equations as

where u is the input, x is the state vector, and y is the output. For single-channel systems, A is an m-by-m matrix where m is the order of the filter, B is a column vector, C is a row vector, and D is a scalar. State-space notation is especially convenient for multichannel systems where input u and output y become vectors, and B, C, and D become matrices.

Taking the z-transform of the state-space equations and combining them shows the equivalence of the state-space and transfer function form:

The Matlab signal-processing function tf2ss converts the transfer function to state-space form.

Example

[A, B, C, D] = tf2ss(b,a)

A =

    -3    -3    -1
     1     0     0
     0     1     0


B =

     1
     0
     0


C =

     2     3     4


D =

     0

The Matlab signal-processing function ss2tf converts from state-space form to transfer function.

Partial Fraction Expansion (Residue Form)

Each transfer function also has a correpsonding partial fraction expansion or residue form representation given by

provided that H(z) has no repeated poles. Here, n is the degree of the denominator a(z). If r is a pole of multiplicity s then H(z) has terms of the form

The Matlab signal-processing function residuez converts transfer function to partial-fraction form.

Example

[r,p,k] = residuez(b,a)

r =

   4.0000 - 0.0000i
  -5.0001 - 0.0000i
   3.0000 + 0.0000i


p =

  -1.0000          
  -1.0000 - 0.0000i
  -1.0000 + 0.0000i


k =

     []

Second-Order Sections

Any transfer function H(z) has a second-order sections representation,

where K is the number of second-order sections that describe the system. Matlab represents the second-order section form of a discrete-time system as an K-by-6 array sos. Each row of sos contains a single second-order section, where the row elements are the three numerator and three denominator coefficients that describe the second-order section:

There are an uncountable number of ways to represent a filter in second-order sections form. Through careful pairing of the pole and zero pairs, ordering of hte sections in the cascade, and multiplicative scaling of the sections, it is possible to reduce quantization noise gain and avoid overflow in some fixed-point filter implementations.

The Matlab signal-processing functions ss2sos and zp2sos perform these operations.

Example

sos = ss2sos(A,B,C,D)

sos =

    0.0061         0         0    1.0000    1.0000         0
  325.9477  488.9216  651.8955    1.0000    2.0000    1.0000


sos(1,1)*sos(2,1)

ans =

     2

Maintained by John Loomis, last updated 14 Oct 1997