Now that we know what
happens when we raw sample a continuous time
complex exponential, we're ready to figure
out the effect of raw sampling on
an arbitrary signal, whether it's band limited or not. So the setup is that we will have a continuous
time signal. For clarity we will use a subscript c to indicate
explicitly continuous time. We raw sample the signal using the sampling period T_s and we obtain a discrete time
sequence X of n, where each sample is equal to the continuous time function computed at multiples of T_s. The question is, given the original spectrum of
the continuous time signal, what is the spectrum of the discrete time sequence
obtained by raw-sampling? The key idea is to exploit
the knowledge that we have gained by examining what happens when we sample
complex exponentials. Let's pick a sampling period T_s and set F_s as 1 over
T_s as per usual, and we pick a complex
exponential whose frequency F0 is less than F_s over 2. So we know that when we raw sample this complex exponential, we will get a discrete time
sequence where the digital frequency Omega
is equal to 2 pi F0 T_s, which is less than pi. Now, if we pick another complex exponential at
a frequency F0 plus F_s, remember F_s is the reciprocal
of the sampling period, and we sample this with the
same sample period as before, we get that the discrete time
complex exponential, we have a frequency
equal to 2 pi, then multiplies F0 plus
F_s multiplied by T_s. So we can split this
into two terms. The first is 2 pi of 0 T_s n, and the second is
2 pi F_s T_s n. Now, F_s and T_n are reciprocals, so this is equal to one. So the second part here it's just simply a multiple of two pi and so we can eliminate it from the phase of the
complex exponential. So what we have found
once again is that, if we sum F_s or
any integer multiple of F_s to the frequency of a complex exponential in continuous time
and we sample it, we will always obtain a
complex exponential in discreet time at
the frequency that is equal to F0 times T_s. Why is this intuition important? Because now if we have the sum of two distinct complex exponentials
whose frequencies are F_s hertz apart and we sample the sum of complex exponentials
with a period T_s, the result will be a complex
exponential whose amplitude is the sum of the amplitudes of
the two distinct components. So this two components there are distinct in continuous time, collapse into the same spectral component
in discrete time. This intuition is important
because we know that, thanks to the Fourier transform, we can always express a continuous time signal
as the combination of an infinite number of
complex exponentials with varying amplitudes. So when we sample a continuous time signal with
a sampling frequency F_s, all the spectral components are F_s hertz apart
will collapse into the same frequency location in the DTFT of the sampled sequence. So back to our task, we want to find out
the spectrum of a raw-sampled sequence starting from the original spectrum of
the continuous time signal. To derive mathematically
the exact shape of the spectrum, we will start by computing the inverse Fourier transform of the original spectrum which is indicated by
the notation X_c of F. We will compute
this inverse Fourier transform at the location
t equal to n T_s. This will give us the value of the continuous time
function in n T_s, which is equal to the sample number n in the
discrete time sequence. Then we will manipulate
the quantity inside this integral until the integral
looks like an inverse DTFT, which will give us the value of the discrete time
sequence in n once again. So since this expression will be equal to this expression, we will be able to derive
the form of the spectrum of the sampled sequence starting from
the original spectrum. As we said before, frequencies there are F_s hertz
apart will be aliased, there will collapse
onto the same location. So it makes sense to split the integral in the inverse Fourier transform into
non-overlapping intervals, we have the
entire frequency axis, and we will compute the
integral independently on sections that
are F_s hertz wide. So this full integral over the entire frequency axis becomes the sum for
k that goes from minus infinity to
plus infinity of the same integral but computed over an interval that goes
from k F_s minus F_s over 2, to k F_s plus F_s over 2. So say for instance k equal to 2, we will be into this integral. This is k F_s, and we will
be computing this portion, so by arranging k over
the entire set of integers, we cover entire frequency axis. Graphically what we're
trying to do is collapse all this frequency
contributions onto the baseband interval between minus F_s over 2 to F_s over 2. So now the value
of X_n is given by this sum of integrals over
this smaller intervals. We now operate
a change of variable. We go from f to f plus k F_s. This corresponds to, instead of integrating over
different sections, we always integrate between
minus F_s over 2 to F_s over 2 but we slide the function to integrate
over this interval. Now, with this change
of variable, we have that e to J 2 pi f
minus k F_s T_s n becomes simply e to the J 2 pi f T_s n because
F_s T_s is equal to 1. So with this change of variable, we move from the
original formulation to this formulation where
the integration limits do not depend on k, and now we can interchange
summation and integral. We have an integral between minus F_s over 2 n F_s over 2 of the sum of multiple copies of the continuous time spectrum, each copy shifted by
a multiple of F_s. This is a periodization of the original analog spectrum
with period F_s. As a matter of fact, we can define this auxiliary function tilde X_c of f. We use the tilde notation to stress the periodicity of this function. This is given by the sum of all possible shifts of X_c
by all multiples of F_s. As we will see later
in more detail, suppose this is
our original spectrum, the periodization with
period F_s will place copies of the spectrum
at all multiples of F_s. With this auxiliary function, the value of Xn becomes the integral over the minus F_s over 2 F_s over
2 interval of the periodized spectrum
times e to the J 2 pi f T_s n. Now, we operate another
change of variable, we set Omega equal to 2 pi f T_s, so that f is equal to
Omega over 2 pi times F_s. So df becomes simply F_s
over 2 pi the Omega. When we replace this
into the integral, we get that the value
of Xn is 1 over 2 pi, times the integral
between minus pi and pi because of the change
of variable, times F_s, times our periodized spectrum computed in Omega
over 2 pi times F_s, times e to the J Omega n d Omega. But this is indeed
an inverse DTFT. This function here that was periodic in F_s in
the previous slide, is now periodic in two pi. So it is indeed
a valid inverse DTFT because the argument is a two pi
periodic function of Omega. So if this is the inverse DTFT
of this quantity, it means that this
is the spectrum of the discrete time
sequence obtained by raw sampling an arbitrary
continuous time function. So once again the spectrum of the raw sampled sequence is equal to the periodized spectrum of the original continuous
time signal scaled so that the minus F_s to
F_s interval is mapped to the minus pi to
pi interval explicitly. This is the sum of an infinite number of copies
of the original spectrum shifted by all
multiples of F_s and re-scaled to fit on the
minus pi pi interval. This is a mouthful in formulas but it's very
intuitive graphically. So let's have a look. Let's take a bandlimited
signal for instance, in this case band-limited
at some frequency F0, and we choose a sampling
frequency so that F_s is larger than two times F0. So larger than twice the maximum positive frequency
of the signal. The auxiliary signal
that we computed before till the X_c of F, the periodized spectrum,
is built by placing a copy of this spectrum here
at all multiples of F_s. So you would have one copy here, one copy here and one copy here, and then by summing
all these copies together. So what we get is this periodic repetition of the fundamental shape
of the spectrum. Now, because we chose a sampling frequency that is larger than the maximum
positive frequency. We don't have any overlap
between this copies and so the periodization doesn't change the shape of the original signal. Then when we want to compute the spectrum of
the raw sampled sequence, all we need to do is
rescale this interval to the minus pi pi interval and
we have the following shape. We can repeat the experiment where now the
sampling frequency is exactly equal to twice the maximum positive
frequency of the signal, this is called critical sampling. When we build
the periodic repetition, we never really build
the periodic repetitions like a way to visualize what's
happening when we sample, the copies touched
but do not overlap. So we still preserve the entire information contained in the spectrum of
the original signal. When we map the minus F_s
over 2 to F_s over 2 interval to the minus pi to pi interval in
the digital domain, once again the shape of
the spectrum is preserved. This is not the case if our sampling frequency is less than twice the
maximum frequency. This is an example where F_s is slightly less than what
should be necessary. So the periodized version
of this spectrum will contain overlap when we put this copies at
all multiples of F_s, the copies overlap and there sum will disrupt the shape of
the original spectrum. Then the digital spectrum
will be still the portion of this periodic spectrum
between minus F_s over 2 and F_s
over 2 map onto minus pi pi, but as you can see the shape is no longer like the original. We could not recover
the original spectrum from the digital spectrum and
this is because of aliasing. Aliasing also occurs if the original continuous time
signal is not bandlimited. In this case no matter how
high the sampling frequency, there will always
be some overlap. If the signal tapers down, we can try and mitigate
the effects of aliasing by choosing a larger
sampling frequency but the repetitions will still
overlap and there will be some distortion introduced
by high-frequency aliasing. In this case you see that the floor of the signal
gets raised because of the tails of this spectral shape
and in continuous time, we will have a spectrum
that will be very close to the original especially in the baseband area
but not quite identical, we will now be able to recover the original spectrum from
the digital spectrum.