Fourier Transform and Signal Spectrum - Digital Signal Processing: A Practitioner's Approach - page 81

0.2

0.25

d

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0 0.2 0.4 0.6 0.8

0 0.2 0.4 0.6 0.8 1

d

Time mapped to frequency

(b)

(a)

Figure 4.3 Output of a block mean filter at an SNR of 0 dB

Figure 4.4 Hardware realisation of a spectrum analyser

Figure 4.4 shows a hardware realisation of the filters in (4.3) to (4.6).
Figure 4.3(a) shows the output y k (4.6) for a specific value of s k . Figure 4.3(b)
shows the distance between the peaks versus the input frequency of the signal s k ;
there is a perfect linear relation. In reality, we don't have to sweep till 0.5 Hz and it
will suffice if we sweep till 0.25 Hz and find the values of y k .

4.2 Discrete Fourier Transform
The DFT relation is derived from the continuous FT of a signal. Consider a signal
x ð t Þ and its FT as

ð 1

x ð t Þ e j2 ft dt :

X ð f Þ , ð 4 : 7 Þ

1

If the signal is such that x ð t Þ¼ 0 for t < 0, then the FT takes the form

ð 1

x ð t Þ e j2 ft dt

X ð f Þ¼ : ð 4 : 8 Þ

0