The sampling theorem gives us a sufficient but not a
necessary condition on the sampling frequency
that we must use when we sample
a continuous time signal. So if we pick a sampling
frequency so that F_s is larger than twice the maximum positive
frequency in the input signal, we are guaranteed an alias
free discrete time sequence. But there are classes of signals, and typically bandpass signals, where the sampling
frequency that is indicated by the sampling
theorem would be way too large with respect
to the amount of information that is actually contained in the
continuous time signal. So a bandpass signal is typically the product of a
modulation of a ration. So you start with
a base band signal like so where the frequency content is say
between minus F_0 and F_0, and it's centered around zero. Then via modulation, you bring this signal up to
a center frequency F_c, which is usually in the higher
regions of the spectrum. Because if you want to transmit this information over
say wireless channel, you need to modulate
it properly so that it occupies a prescribed portion
of the spectrum. So the actual information
content in the signal has an effective bandwidth
of two times F_0, and F_0 usually is way, way smaller than F_c. However, a literal interpretation of the sampling
theorem tells you that you have to sample at least
twice the maximum frequency, which in this case
would be F_c plus F_0. So very, very large sampling frequency especially
with respect to F_0. So we could try
a naive approach and say well, since the effective bandwidth
is just two times F_0, I'm going to just sample at two times F_0 and
see what happens. In most cases, what will
happen is that you will get a lot of copies of
the spectrum because, of course, now, the periodization frequency
is very small. You will get overlap which will destroy the information contained
in the bandpass signal. But if you play around with
the sampling frequency while remaining in
the low frequency range, so smaller than F_c, you can probably find
some lucky numbers, and you will get a situation
in which the alias copies, instead of overlapping, actually interleave neatly without
stepping onto each other. In this case, the
information contained in the bandpass signal is preserved and the sampling operation goes under the name of
bandpass sampling. Let's now look at the conditions under which you can perform bandpass sampling and what is the result of the
sampling operation. So in order to use
bandpass sampling, first of all, you have to have an input signal
of this bandpass. We indicate this by saying that the spectrum of the signal
has to be zero almost everywhere except for a region centered around
the modulation frequency F_c. The width of this region
is two times F_0. So this is the support
of the signal on the positive frequency axis,
and of course, this is symmetric if
the signal is real with a similar region centered
around minus F_c. So intuitively, in
order to preserve the information contained
in the bandpass signal, your sampling frequency cannot be less than two times F_0. If your signal was baseband,
centered around zero, so between minus F_0 and F_0, then the sampling frequency
would have to be twice F_0. So there's no escaping that. Now, the second condition
that allows you to perform bandpass sampling is called the baseband condition. This is the condition that
will allow the copies to interleave rather
than overlap, and it is expressed by
the requirement that the sampling frequency should be a submultiple of
twice the modulation frequency. In other words F_s
should be equal to two times F_c divided by K
where K is some integer. So when this conditions
are fulfilled, we choose a sampling period
T_s equal to one over F_s, and the resulting sequence
of samples x of nT_s will be an alias free baseband
discrete time version of the passband signal. So we have both sampling and demodulation to
baseband in one go. Let's look again at
the process graphically. So here's your
baseband signal with effective bandwidth two times
F_0 and modulated to F_c. Here, we choose some parameter K, may be I don't know
equal to five, and we choose a sampling
frequency which is equal to two times F_c over five. This will allow
this petrel copies to interleave instead
of overlapping. You see that in the baseband, you have one repetition that is been brought
back to baseband. This is the intermediate signal that we use just to
understand what happens during sampling but
the actual DTFT of the sampled sequence is this one where we bring this part to the
minus Pi Pi interval, and you see, it is
a demodulated version of the passband signal. So let's look at
a practical example. The AM radio band is a portion of the frequency spectrum that is located for
AM radio transmission, and it usually goes, it
depends on the country, but usually between
500 kilohertz and 1.6 megahertz. The portion of the spectrum
is divided into independent channels
that are then allocated to different
radio stations, and the usual channel width
is nine kilohertz. So the F_0, half the effective bandwidth of a channel is 4.5 kilohertz. Let's pick a channel with the center frequency
of 1.5 megahertz. So pretty much at
the end of the AM band. In theory, if we wanted
to sample the information contained into that radio channel using the sampling theorem, we'd have to use a sampling
frequency which is two times the highest frequency
contained in the channel. So two times 1,504,500 hertz for a sampling period of about one microseconds
which is pretty fast. The antialias condition for
the single channel, however, imposes a much lower
sampling frequency if the effective channel width
is nine kilohertz, then the sampling frequency is equal to the
effective channel width. So we just need to use a
sampling frequency that is higher than nine kilohertz to preserve the information. The baseband condition on the other hand
requires us to pick a sampling frequency so that we can express the
modulation frequency. So the center band frequency has a multiple of half the
sampling frequency. In other words, kF_s over two must be equal to 1.5 megahertz. We can pick
any integer as long as the resulting sampling frequency fulfills the antialias condition. So let's pick k equal to 300. With that, we have a
sampling frequency of 10 kilohertz for a sampling period of
about 0.1 microseconds. So you have a savings of about two orders of magnitude
in the sampling speed.