Digital Signal Processing Reference
In-Depth Information
Example 4.23.
Determine the real coefficients of the second order IIR that results from using the two
complex conjugate poles
[
0
.
65
+
j
0
.
65
,
0
.
65
−
j
0
.
65
]
We convolve the coefficient representations of each:
y = conv([1, -(0.672 +j*0.672)],[1, -(0.672 - j*0.672)])
which yields
y = [1, -1.344, 0.9034]
π/
4 radians, we would expect the peak response of the IIR to lie
at one-quarter of the Nyquist rate, or one-eighth of the sample rate. We can explore this with a script that
receives one pole as its magnitude and angle in radians, computes the complex conjugate pole, computes
the net real coefficients, and filters a linear chirp-a call that yields the answer for the specific problem at
hand is
Since the poles have angles of
±
LVRealFiltfromCCPoles(0.95, pi/4)
which results in Fig. 4.39.
function LVRealFiltfromCCPoles(PoleMag,PoleAng)
% LVRealFiltfromCCPoles(0.95,pi/4)
Pole = PoleMag*exp(j*PoleAng); cPole = conj(Pole);
rcoeffs = conv([1, -(Pole)],[1, -(cPole)]); SR = 1024;
t = 0:1/(SR-1):1;x=chirp(t,0,1,SR/2);
y = filter([1],[rcoeffs],x); figure(8); plot(y);
xlabel('Sample'); ylabel('Amplitude')
The results above will be again demonstrated and generalized when the topic of the
z
-transform
is taken up in Volume II of the series. We will also explore different filter topologies or implementations
for a given net impulse response.
The LabVIEW VI
DemoDragPolesVI
allows you to select as a test signal a unit impulse, a unit step, or a chirp. The pole or complex conjugate
pair of poles is specified by dragging a cursor in the
z
-plane. From these poles, the VI forms an IIR and
filters the selected test signal. The real and imaginary outputs of the filter are plotted. The importance
of using poles in complex conjugate pairs can readily be seen by alternately selecting “Single Pole” and
“Complex Conjugate Pair” in the Mode Select box. The use of a complex conjugate pair of poles results
in all-real filter coefficients and an all-real response to an all-real input signal.
A script that allows you to move the cursor in the complex plane and see the frequency and impulse
responses arising from a single pole or a complex-conjugate pair of poles is
ML
_
DragPoleZero
