Fourier Transform and Signal Spectrum - Digital Signal Processing: A Practitioner's Approach - page 99

1.4

0

1.2

Experimental data

True data

− 50

1

0.8

− 100

0.6

0.4

−150

0.2

−200

0

0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4

(a) (b)

Figure 4.18 Frequency domain least-squares data

Figure 4.18 shows values of A i

i versus frequency at an SNR of 0 dB for
the system defined by (4.44) and (4.45). True values are superimposed on the
computed values. Each true value is obtained by sampling H ð z Þ on the unit circle at
the desired frequency. Note that the values of A i

and

i are very accurate even
when the SNR is 0 dB. This is the primary reason why the estimates are noise
tolerant.

and

4.9.2 Estimating the Parameter Vector
Using the above data and (4.43), we get the solution as

p ¼ 0 ð : 9008 0 : 4055 0 : 1312 0 : 2502 0 : 1270 Þ:

The actual values can be obtained by looking at (4.44) and (4.45):

p¼ 0 ð : 8994 0 : 4045 0 : 1260 0 : 2520 0 : 1260 Þ:

4.10 Summary
In this chapter we used a matrix approach to explain the principles of the FFT.
We have not used butterfly diagrams. Circular convolution was explained in
depth. We presented a hardware scheme for implementation in real time. We
looked at a problem of estimating frequency using a DFT. We elaborated a
hardware structure for real-time implementation of continuous spectrum update.
We ended by covering the principles of the network analyser, an example from
RF systems.