where * denotes the linear convolution operation and the operation defined in (3.2.12) is
called the convolution sum. The input signal, xn, is convoluted with the impulse
response, hn, in order to yield the output, yn. We will discuss the computation of
linear convolution in detail in Chapter 5.
As shown in (3.2.12), the I/O description of a DSP system consists of mathematical
expressions, which define the relationship between the input and output signals. The
exact internal structure of the system is either unknown or ignored. The only way to
interact with the system is by using its input and output terminals as shown in Figure
3.8. The system is assumed to be a `black box'. This block diagram representation is a
very effective way to depict complicated DSP systems.
A digital system is called the causal system if and only if
hn 0, n < 0:
A causal system is one that does not provide a response prior to input application. For a
causal system, the limits on the summation of the Equation (3.2.12) can be modified to
reflect this restriction as
hkxn À k:
Thus the output signal yn of a causal system at time n depends only on present and
past input signals, and does not depend on future input signals.
Consider a causal system that has a finite impulse response of length L. That is,
n < 0
0 n L À 1
n ! L .
Substituting this equation into (3.2.14), the output signal can be expressed identically to
the Equation (3.1.16). Therefore the FIR filter output can be calculated as the input
sequence convolutes with the coefficients (or impulse response) of the filter.
A digital filter can be classified as either an FIR filter or an infinite impulse response
(IIR) filter, depending on whether or not the impulse response of the filter is of
finite or infinite duration. Consider the I/O difference equation of the digital system
yn bxn À ayn À 1,
where each output signal yn is dependent on the current input signal xn and the
previous output signal yn À 1. Assuming that the system is causal, i.e., yn 0 for
n < 0 and let xn dn. The output signals yn are computed as
DSP FUNDAMENTALS AND IMPLEMENTATION CONSIDERATIONS