So we have seen that
sinc interpolation is the ideal
interpolation method, in the sense that it gives us a maximally smooth
continuous time function that goes through the set of data points that constitute
the discrete time signal. So the ingredients for sinc interpolation is
a discrete time signal, x of n, which we'll also have a frequency domain representation
by way of its DTFT, X of e to j omega. Then we choose an interpolation interval T_s that gives us the spacing in seconds between samples and
the sinc function. With this three ingredients, we get the smooth
continuous time signal x of t, where t now is a real variable that
indicates time in seconds. The question is, what is the spectrum of
this continuous time signal? The intuition is that
given that the sinc is the prototypical
band limited function, the spectrum of
the interpolated function will be itself band limited. So let's formalize
this intuition. If we have a sinc with
a certain time-based, this is a sinc function
that will be equal to 0 for all multiples of T_s. In the frequency domain, its frequency representation
is erect of support F_s, where F_s is one over T_s. So this is the sinc, it will go to zero at
all multiples of T_s, and in the frequency domain, we have a support of F_s hertz from minus F_s
over 2 to F_s over 2, with F_s equal to 1 over T_s. The continuous time signal is given by the
interpolation formula. Again, mixed domain convolution where we have
the discrete time samples here and shifted versions of the scale sinc
prototype function. If we tried to compute the frequency domain
representation of this function, so the Fourier Transform in continuous time of
this continuous time signal, we apply the definition
of the Fourier Transform, the integral from
minus infinity plus infinity of our
continuous time function, times e to the minus
j two pi ft in dt. Now, we replace
the expression for xt with the mixed
domain convolution. Now, we have a summation
and an integral, and if we assume that
all signals have finite energy, we can interchange their order. We have now an outer sum that indexes the
discrete-time samples, times this expression here, which is nothing but the Fourier Transform
of the sinc function. Now, this sinc function is scaled by T_s and delayed by nT_s. So its Fourier Transform
will be a wracked, as we've seen before, times a phase factor that
takes into account the shift. This phase factor is e to
the minus j two pi f nT_s. Now, this part here
does not depend on n, so we can take it
out of the sum and we replace one over F_s by T_s, we multiply this by
rect f over F_s, and then we're left with
this summation here, which is nothing but the DTFT of our original discrete-time
signal computed in omega equal to two
pi times f over F_s. So in summary,
the Fourier Transform of the continuous time signal
is equal to 0 outside of the minus F_s over 2 to F_s over 2 interval
because of this rect term. This guy would be
two pi periodic, and this guy kills the repetitions outside of
the base band interval. Inside the base band interval, between minus F_s over
2 and F_s over 2, we have a scaled
version of the DTFT of the original
discrete time signal between minus pi and pi. Graphically, it looks like so, assume this is the DTFT
of the signal, we just represent the portion
between minus pi to pi. If we now interpolate the signal with a given value for T_s, we have that the spectrum outside of the minus F_s over 2, F_s over two interval
is going to be 0,, and inside that interval
is going to be a scaled version of the original
discrete time spectrum. Now, the interesting thing
is that if we pick an interpolation interval
that is bigger, so we're letting
more time elapse between samples where interpolating
the signal more slowly, the shape of the spectrum
will be the same, but it will be contracted
closer around zero. Because now, 1 over F_s over 2 with a larger period
will be smaller. Conversely, if we choose an interpolation interval
which is smaller, so we interpolate much faster, the spectrum will be the same, but it will be stretched out to a larger
frequency support. So to sum up our findings, given a discrete time signal, we pick an interpolation
period T_s, we perform sinc interpolation. The resulting continuous
time signal will have a spectrum which is F_s band-limited with
F_s equal to 1 over T_s. If we perform
a fast interpolation, which means our interpolation
interval is small, we will have a wider spectrum. If we perform
a slow interpolation, the spectrum will be narrower. This starting from
the same discrete-time data. For those who remember,
this is like the changing the speed
of a record player. If you play the record faster, which corresponds with
faster interpolator, you have a chipmunk voice. It will sound like this.
This means that there is more energy at
the high frequencies. On the other hand,
if you play a record slower with respect
to its nominal rpm, you will have a very
low voice like this. Namely, the spectrum will contain mostly low frequencies which is consistent with
a slow interpolation interval.