## 11.3 STATE VARIABLE DESCRIPTION OF DIGITAL FILTERS

Consider the signal flow graph (SFG) of an *N*-th order digital filter in Fig. 11.5. We can represent this filter in the following recursive matrix form,

The boldfaced letters imply a vector or a matrix. In the above representation **x** is the state vector, *u* is the input, and *y* is the output of the filter; **x**, **b**, and **c** are *N* × 1 column vectors; **A** is *N* × *N* matrix; *d*, *u* and *y* are scalars. Let **f**_{i}(*n*) be the unit-sample response from the input *u*(*n*) to the state **x**_{i}(*n*) and let **g**_{i}(*n*) be the unit-sample response from the state **x**_{i}(*n*) to the output **y**_{i}(*n*). To avoid internal overflow, it is necessary to scale the inputs to multipliers. Since the signals **x**(*n*) are inputs to multipliers in Fig. 11.5, we need to compute **f**(*n*) for scaling. Conversely, to find the noise variance at the output, it is necessary to find the unit-sample response from the location of the noise source *e*(*n*) to *y*(*n*). Thus **g**(*n*) represents the unit-sample response of the noise transfer function except for a delay, which has no effect on the output noise variance.

*Fig. 11.6* Signal flow graph of **g**(*n*).

From the SFG of Fig. 11.5 we can write,

Then we can write the *z*-transform of ...