Question

1. (a) What is the Fourier transform of a constant amplitude (A), constant frequency (ωo) RF pulse of duration T: x(t) = Acos(ωot) for t over –T/2 < t < T/2?

(b) Sketch the amplitude spectrum for two durations, T and 2T and describe how the spectrum changes.

(c) What does this say about the accuracy of frequency measurement and the associated observation window?

Answer 1

Solution:

(a) Given that

The fourier transform of x(t) is

(b)

Amplitude Spectrum:

(c)

The accuracy of frequency measurement can be calculated from the short-time fourier transform

w(t) is the window employed

Suppose Blockman window in adopted in the STFT, it can be expressed as

L is the window length i=0,1----L-1

The frequency measurement based on FFT is not suitable for dynamic frequency measurement. Frequency measurement based on wavelet transform better than the frequency measurement of sampling time.

The frequency measurement based STFT given better accuracy compare to the frequency measurement based on FFT and wavelet transformer

The accuracy of frequency measurement based on short time fourier transformer given the better result.

Note: Please comment below if you have any doubts.