Digital Signal Processing Reference
In-Depth Information
0.7
0.7
0.6
0.6
0.5
0.5
Third-order
fit
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.25
0.3
0.35
0.4
0.45
0
0.1
0.2
0.3
0.4
0.5
Frequency
Frequency
(a)
(b)
Figure 4.16 DFT of assembled records and curve fitting
For applications of this type it is best to take the DFT approach. Figure 4.16(a)
shows the spectrum of the signal with a conspicuous maximum around 0.3 Hz. We
have fitted a third-order polynomial around this maximum. For this polynomial we
have obtained the maximum very accurately by computing the values for any
desired frequency. This allows us to choose the resolution, leading to an accurate
positional estimate
5
of the peak, as shown in Figure 4.16(b).
Often a simple DFT computation may not suffice on its own and we need to do
more manipulation to obtain better results. In the case of heterodyning spectrum
analysers, similar curve fitting is performed at the output of the bandpass filter.
4.9 Parametric Spectrum in RF Systems
Conventional methods cannot be applied for RF systems. Impulse response is
irrelevant and these systems can only be characterised using s-parameters in the
S
matrix. However, the notion of a transfer function exists and we can make
measurements of amplitude and phase.
Frequency domain least squares (FDLS) is useful for obtaining the transfer
function of the RF device. In this approach, the system under consideration is
excited by fixed frequencies and output amplitude and phase are measured at steady
state. Suppose we obtain the following measurements using the experimental set-up
in Figure 4.17. This set-up is typical of a network analyser. The staircase is used so
the system can settle down and reach steady state for measurements.
Consider the unknown device to have a model given as
y
k
¼
a
1
y
k
1
þ
a
2
y
k
2
þ
b
1
u
k
þ
b
2
u
k
1
þ
b
2
u
k
2
:
ð
4
:
35
Þ
5
Location of the maximum rather than the value of maximum.
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