So in the previous lectures we
have studied the space of FS band-limited functions,
namely functions whose spectral support is contained between
-Fs over 2 and Fs over 2. The set of such functions form
a Hilbert space and we have found an orthogonal basis for the space,
which is composed of shifted versions of the sinc function scaled by the
parameter Ts which is equal to 1 over Fs. So once we're given
a Fs-band limited function, we can compute its basis decomposition and the analysis coefficients are the inner
product between R function and all the basis vectors and we have seen
that for FS-band limited functions. These expansion coefficients
are equal to simply a scaled version of the original function
computed at multiples of Ts. Now this inner product,
we can also look at it as a convolution between a sinc scaled by Ts and
our original function computed in nTs. These two formulations
are exactly the same. But now with this kind of interpretation,
we can see that this is equivalent to a filtering operation
where we have an ideal filter here and ideal low-pass filter with cutoff
frequency Fs over 2 and then we have a a raw sampler that extracts the values
of this convolution at multiples of Ts. Now since the original
function is Fs-band limited, the effect of this filtering
operation are irrelevant. So we can eliminate this filter
in this block diagram and just perform a raw sampling of
the original band limited signal and we will obtain exactly the same expansion
coefficients as if you had performed a full inner product between
the signal and the basis vectors. Which by the way is very convenient since
we know that we can never implement exactly a filter with a sink impulse
response, even in continuous time. So the temptation is to
always use just raw sampling. In other words,
can we take a continuous time signal? Take regularly spaced samples at
multiples of a given sample period and assume that, that is a full
representation of the original continuous time signal without any
loss of information. Well, we already know
the answer to be negative. You remember the wagonwheel effect
example, which we introduced when we talked about the maximum angular speed
of a discrete time complex exponential. The effect can be explained by considering
that the movie is a sequence of frames, namely a sequence of snapshots of
the continuous time movement of the wheel. And since the wheel at one point is
moving too fast with respect to the raw sampling frequency, we have this
weird visual artifact where the wheel seems to be moving backwards with
respect to the movement of the carriage. We will now study this
phenomenon more detail when applied to the sampling of
non band limited functions or sampling band limited functions with
a sampling rate that is not small enough.