DSP Technology 1167tems than the Fourier transform. In addition, the
analysis of systems is easier due to the convenient nota-
tion of the Z-transform.^1 The Fourier transform is
defined as
while the Z-transform is defined as
When working with linear time invariant systems, an
important relationship is that the Z-transform of the
convolution of two sequences is equal to the multiplica-
tion of the Z-transforms of the two sequences,
i.e., H(z) is
referred to as the system function (a generalization of
the transfer function from Fourier analysis).
A common use of the Z domain representation is to
analyze a class of systems that are defined as linear
constant-coefficient difference equations that have the
form of
(31-13)where,
the coefficients ak and bk are constant (hence the name
constant coefficient).
This general difference equation forms the basis for
both finite impulse response (FIR) linear filters, and
infinite impulse response (IIR) linear filters. Both FIR
and IIR filters are used to implement frequency selective
filters (e.g., high-pass, low-pass, bandpass, bandstop,
and parametric filters) and other more complicated
systems.
FIR filters are a special case of Eq. 31-13, where
except for the first coefficient, all the ak are set to 0,
leading to the equation(31-14)The important fact to notice is that each output
sample y[n] in the FIR filter is formed by multiplying
the sequence of coefficients (also known as filter taps)by the input sequence values. There is no feedback in an
FIR filter—i.e., previous output values are not used to
compute new output values. A block diagram of this is
shown in Fig. 31-8 where the z–1 blocks are used to
denote a signal delay of one sample (i.e., the Z-trans-
form of the system h[n]=G>n 1 @).
An IIR filter contains feedback in the computation of
the output y[n]—i.e., previous output values are used to
create current output values. Because of this feedback,
IIR filters can be created that have a better frequency
response (i.e., steeper slope for attenuating signals
outside the band of interest) than FIR filters for a given
amount of computation. However, most DSP architec-
tures are optimized for computing FIR filters—i.e.,
multiplying and adding signals together continu-
ously—so the choice of which filter style to use will
depend on the particular application.31.6 Sampling of Continuous-Time SignalsThe most common way to generate a digital sequence is
to start with a continuous-time (analog) signal and cre-
ate a discrete-time signal. For example, speech signals
are continuous-time signals because they are continuous
waves of acoustic pressure. A microphone is the trans-
ducer that converts the acoustic signal into a continu-
ous-time electric signal. In order to process this signal
digitally, it is necessary to convert this signal into the
digital domain. Finally, after processing, it is often nec-
essary to convert the discrete-time signal back into a
continuous-time signal for playback through a loud-
speaker system.
The process of converting an analog signal to a
digital signal is often be modeled as a two-step process,
as shown in Fig. 31-9, of converting a continuous-time
signal to a discrete-time signal (with infinite resolution
of the amplitude) and then quantizing the discrete-time
signal into finite precision values (creating the digital
sequence) that can be processed by a computer.^1 The
process of converting the continuous-time signal into a
discrete-time signal will be introduced, and then quanti-
zation will be reviewed. The quantization step is neces-
sary to create a sample value that has a data word size
that is compatible with the arithmetic capabilities of the
target DSP. All real-world analog-to-digital converters
(A/Ds) perform both the sampling and quantization
process internal to the A/D device, but it is useful to
discuss the subsystems separately because they have
different significance and design trade-offs.Xe jZ xk>@e–jZk
k=¦
Xz xk>@z–k
k=¦
yn>@=xn>@^ hn>@Yz =Xz Hz.akyn k>@–
k 0=N¦ bkxn k>@–
k 0=M= ¦
yn>@ bkxn k>@–
k 0=M