So we have seen that for
band-limited signals, the projection onto the
orthogonal basis for the space of bandlimited functions is
equivalent to raw sampling. On the other hand, if we raw sample
an arbitrary signal, we may incur aliasing if
the sampling frequency is not high enough or if
the signal is not bandlimited. So given the sampling
frequency F_s, which may be fixed because of the hardware that we
use to do the sampling, what are our sampling strategies? Well, if the signal happens to be bandlimited to F_s
over 2 or less, so the maximum
positive frequency of the input signal is less than half the
sampling frequency, then raw sampling is fine. It is equivalent to sinc
sampling and we know that this preserves the information
contained in the analog signal. If the signal is not bandlimited
or if it is bandlimited but the maximum frequency
is higher than F_s over 2, then we have two choices. We can bandlimit the signal in the continuous time
domain prior to sampling. This is achieved with
an ideal low-pass filter with cutoff frequency F_s over 2, and it is really
equivalent to projecting the signal onto the space of
F_s band-limited functions. If we forego this
filtering operation, then we can still raw
sample the signal, but we will incur aliasing. Now, we have seen that aliasing introduces error that we
cannot really control. The overlap between the copies
of the spectrum has very disrupting effects on the overall shape of the
spectrum of the sampled signal. So the sensible choice
is to go for an arrow that we can
actually control and nil, and so we will
bandlimit the signal in continuous time before
raw sampling it. But badlimiting is also
optimal because, as we said, corresponds to a projection onto the space of
bandlimited functions, so it minimizes
the mean square error between the sampled signal
and the original signal. So in all cases, the correct way to sample a signal is to
perform a full basis expansion uses in the sinc basis for the space of F_s
bandlimited functions. Now, we know that
this expansion in the case of a bandlimited signal will
correspond to raw sampling, but in all other cases
will correspond to a projection over that space. Once we have the
expansion coefficients, we can reconstruct
an approximation to the original
continuous time signal using the sinc
interpolation formula, and again, in the case
of a bandlimited signal, this will be identical to the original continuous
time signal otherwise, it will be an
approximation that is optimal in the mean square sense. The chain of sampling
and reconstruction can be described
schematically like so. The projection onto
the sinc basis corresponds to an ideal low-pass
filtering operation in the continuous time domain
where the filter is an ideal low-pass with
cutoff frequency F_s over 2. Then we have raw sampling, and then we have sinc
interpolation using a sinc kernel. From the original
continuous time signal, we end up with an approximation that will be either equal
to the original signal if the signal is bandlimited or will be optimal in
the mean square error sense. Using the power of
vector space abstraction, we can visualize
this schematically like so. Imagine that the space of bandlimited function is
a one-dimensional vector here, and now we have a function
that is not bandlimited and therefore is not
collinear to this vector. Once we project this function onto the space of
bandlimited functions, we are building
an approximation that is given by a linear combination
of the basis vector for BL weighed by the analysis coefficients that we computed by an orthogonal projection. This approximation, in red here, is the approximation that
minimizes the error namely the distance between x and the space of
bandlimited functions. So if we go back to the case of a non bandlimited signal and we choose
a sampling frequency F_s, what projection does is perform an ideal
filtering and cut off the parts of the signal
that are not within the minus F_s over 2 to
F_s over 2 interval. At this point,
the periodic repetition of the spectrum will not overlap. We will have lost
some information, but in a controlled fashion, and then the discrete time
spectrum will be exactly equal to the portion of the original spectrum between minus F_s over 2
and F_s over two. So that if we tried
to reconstruct the continuous time signal
from the samples, we will get exactly a copy of the filtered original version.