Lecture notes for question 3

(Related concepts show the approach for the problem three)

In this diagram both the magnitude ( triangle) and phase (smooth curve) of are shown for convenience. Note that the phase function ends abruptly at because when the magnitude of a complex number is zero the phase is meaningless.

According to the sampling theorem proved below, if is bandlimited and if is small enough, then knowledge of the samples alone (shown below) is enough to completely and exactly reconstruct the entire continuous time signal.

The Poisson Sum Formula:

If , then given any real number, , the following identity holds

. (8.2)

To illustrate this result, consider a bandlimited signal, , with spectrum shown below.

The aliased version shown above is drawn assuming that . If this were not the case, then would contain distortions due to overlapping images, as the student can readily see.

First proof

The constants, , are obtained as

Substituting these values of into the fourier expansion for yields

.

is a periodic sequence of unit impulses and can be expressed as

.

Substituting this into the expansion for yields

. (8.4)

The right hand equality is equivalent to the Poisson Sum Formula. Q.E.D.

We define the ideal sampled version of by

RECOVERING THE ANALOG SIGNAL FROM ITS SAMPLES

Using equation (8.8) it is easy to show how to recover from its ideal sampled version, . The diagram below illustrates the reconstruction procedure.

It is clear by taking the product, , that the output of the reconstruction filter in the diagram above will have the spectrum shown below.

When this signal is passed through the reconstruction filter the output, by linearity and time invariance, is clearly given by

(8.10)

Finally, by viewing the reconstruction process in the frequency domain we have already shown that the reconstruction filter output is , so that we now have the Nyquist Reconstruction Formula

. (8.12)