1) A band limited signal of finite energy , which has no frequency components higher than W hertz , is completely described by specifying the values of the signal at instants of time separated by 1/2W seconds and
2) A band limited signal of finite energy, which has no frequency components higher than W hertz , may be completely recovered from the knowledge of its samples taken at the rate of 2W samples per second.
There are two parts :
I)Representation of g(t) in terms of its samples
II)Reconstruction of g(t) from its samples
PART I: Representation of g(t) in its samples g(nTs)
Let g(t) be a bandlimited signal whose bandwidth is fm i.e (ωm = 2πfm). δT (t) is the sampling signal with fs = 1/T > 2fm.
• Let gs(t) be the sampled signal. Its Fourier Transform Gs(ω) is given by
• If ωs = 2ωm, i.e., T = 1/2fm. Therefore, Gs(ω) is given by
To reconstruct g(t), you must recover input signal spectrum G(ω) from sampled signal spectrum Gs(ω), which is possible when there is no overlapping between the cycles of Gs(ω).
Possibility of sampled frequency spectrum with different conditions is given by the following diagrams:
II) Reconstruction of g(t) from its samples
Step 1 : Take inverse Fourier transform of Gs(w).
Step 2 : Show that g(t) is obtained back with the help of interpolation function.