Digital Signal Processing Reference
In-Depth Information
This is the superposition property, where an overall output response is the sum of
the individual responses to each input. Shift invariance implies that if the input is
delayed by
m
samples, the output response will also be delayed by
m
samples, or
x
(
n
-
m
)
Æ
y
(
n
-
m
). If the input is a unit impulse
d
(
n
), the resulting output response
is
h
(
n
), or
d
(
n
)
Æ
h
(
n
), and
h
(
n
) is designated as the impulse response. A delayed
impulse
d
(
n
-
m
) yields the output response
h
(
n
-
m
) by the shift-invariance
property.
Furthermore, if this impulse is multiplied by
x
(
m
), then
x
(
m
)
d
(
n
-
m
)
Æ
x
(
m
)
h
(
n
-
m
). Using (4.20), the response becomes
•
Â
()
=
()
(
)
(4.21)
yn
xmhn m
-
=-
•
m
which represents a convolution equation. For a causal system, (4.21) becomes
•
Â
()
=
()
(
)
(4.22)
yn
xmhn m
-
=-
•
m
Letting
k
=
n
-
m
in (4.22) yields
•
Â
0
()
=
()
(
)
(4.23)
yn
hkxn k
-
k
=
4.3 FIR FILTERS
Filtering is one of the most useful signal processing operations [1-47]. DSp are now
available to implement digital filters in real time. The TMS320C6x instruction set
and architecture makes it well suited for such filtering operations. An analog filter
operates on continuous signals and is typically realized with discrete components
such as operational amplifiers, resistors, and capacitors. However, a digital filter, such
as an FIR filter, operates on discrete-time signals and can be implemented with a
DSp such as the TMS320C6x. This involves use of an ADC to capture an external
input signal, processing the input samples, and sending the resulting output through
a DAC.
Within the last few years, the cost of DSp has been reduced significantly, which
adds to the numerous advantages that digital filters have over their analog coun-
terparts. These include higher reliability, accuracy, and less sensitivity to tempera-
ture and aging. Stringent magnitude and phase characteristics can be achieved with
a digital filter. Filter characteristics such as center frequency, bandwidth, and filter
type can readily be modified. A number of tools are available to design and imple-
ment within a few minutes an FIR filter in real time using the TMS320C6x-based
DSK. The filter design consists of the approximation of a transfer function with a
resulting set of coefficients.
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