So let's go back to
our original question. We know that we can convert any discreet time sequence into a continuous time signal
via sinc interpolation, and when we do that we obtain a signal that
is bandlimited. We have chosen T_s equal to 1. So given discrete time sequence, we will get a one
bandlimited signal. Now the question is does
any one bandlimited signal admit a discrete time
representation that doesn't lose any of the information of
the continuous time signal? We have found that the space of one bandlimited functions is a Hilbert space and we have found an orthonormal basis
for this space. This is accountable basis, and it will turn out that the discrete time
representation of any member of the space of
one bandlimited functions is the set of analysis
coefficients when we perform basis decomposition of the signal onto the orthonormal basis
that we have just found. Let's work this out in detail. So let's pick
any continuous time signal that is one bandlimited, let's consider the
orthonormal basis composed of sheets of the sinc function that we found in the previous module, and we compute
the decomposition of this element onto
this orthonormal basis by taking the inner product between each element of the basis
and the signal itself. So we write this as
the inner product between sinc of t minus n, this is the basis
vector number n, and x of t are signal because of the symmetry of the sinc we can flip the sign of
the argument of the sinc, and we obtain that
this inner product is once again the convolution between a sinc and our bandlimited signal. To compute this convolution, we compute the inverse transform of the product of the transforms. So we have the integral over the entire frequency axis of
the product between rect of f and the fourier transform of the original signal
times e_j2Pifn. Now X of f is one bandlimited, and the rect function is equal to 1 over the minus the 1f interval. So it doesn't affect the
integral and we're left with the inverse fourier transform
of the fourier transform of the input computed in n. So that the expansion
coefficient, the inner product between
the basis vector number n, and the input is simply
the value of the input computed for t equal
to n. In other words, the coefficients of
the basis expansion are the samples of
the continuous time signal. So the analysis formula
says that we can decompose any one
bandlimited function onto the orthonormal basis represented by shifted
versions of the sinc, and the analysis
coefficients will be the samples of the function. The synthesis formula
says that we can go back from the set of analysis coefficients to the original continuous
time signal, and therefore a one
hertz bandlimited signal and a set of samples of the signals spaced
one second apart are equivalent to representations of the same information. In the general case of an
F_s bandlimited function, the analysis formula
remains the same. The basis in this case will be shifted versions of
a scaled sinc function, and the analysis
coefficients will be samples of the
continuous time signal at multiples of T_s, and scaled by T_s they're just a normalization
factor in front. But once again, we
have that a set of regularly spaced
sample T_s seconds apart fully characterize
the continuous time signal. With this samples, we can use the sinc interpolation formula to go back to the continuous
time domain as we saw before. So let's review the philosophy behind the sampling theorem. We recognize that
this space of f_s bandlimited functions
is a Hilbert space, and we set a time-based T_s
equal to one over F_s, and with this we build an orthogonal basis for
this space by taking all possible shifts
at multiples of T_s of a scaled sinc
function scaled by T_s. Now, for any function that belongs to the space of
F_s bandlimited functions, the coefficients when you perform a basis
expansion are simply the scaled values of the function computed
all multiples of T_s. So for any function that
is F_s bandlimited, a sufficient representation
is simply the sequence of regularly spaced
samples x of nT_s. As a corollary, if you have a function that
is F_s bandlimited, this function will
be bandlimited for all F that are larger than F_s. So for any function that
is F_s bandlimited, a sufficient representation
is the sequence of samples x of nT_s for any T_s that
is less than one over F_s. So the maximum frequency
in the signal determines the maximum period that we can use for sampling, but we can always use
a smaller period. Let's look at it from
another angle again. We take a continuous time signal, we look at its maximum
positive frequency component, let's call this F_N. We can sample this signal
with no loss of information, as long as we use a sampling
frequency that is at least twice as large as the maximum positive
frequency of the signal, or equivalently using
the sampling period that is less than one over two times the maximum
positive frequency. F_N is often called the Nyquist
frequency of the signal, and two times F_N is the minimum sampling frequency
that we must use, If you don't want to incur
any information loss.