Hi and welcome to Module 9.4
of Digital Signal Processing.
In the previous module we saw an
interesting signalling scheme that allow
us to increase the data rate while
keeping the same probability of error for
a given power constraint.
The problem is that the efficient
communication alphabet that we devised
is complex valued,
where as we know that physical
channels can only handle real values.
So in this module we'll
see how to transmit and
recover a complex valued simple
stream over a real value channel.
Then we will follow this with a concrete
design example for the telephone channel.
And finally we will compare the
performance of the system to the ultimate
limit in data rate that is given
us by channel's capacity formula.
So let's review where we stand
in terms of transmitter design.
We have the user's bitstream
that comes into the system.
This is sent through a scrambler
that makes sure that the resulting
bitstream is equiprobable.
The mapper will split
the bitstream into m bit chunks,
any chunk will be associated
to a complex valued symbol.
This will create a complex
value sequence a[n].
And to fit that over the bandwidth
prescribed by the channel
we have to upsample it,
which means inserting K minus one zeros
after each symbol of the sequence.
And then low passing the sequence with
a filter with cutoff frequency pi over K.
Now the sequence b[n],
the upsample sequence that now fits
the bandwidth constraint,
is a complex valued baseband signal.
Graphically we can show this.
If this is the bandwidth constraint
of the channel, positive and
negative frequencies.
The sequence B of N,
is a baseband sequence whose bandwidth
fits the bandwidth
prescribed by the channel.
Now the question is,
how do we modulate this while also
making the transmitted signal real?
And making it real in a way
that we can retrieve it and
reconstruct the baseband
complex signal at the receiver.
So the way it's done is
very straight forward.
We just take the real part of the baseband
signal, multiplied by e to the j omega c
n where omega c is the center frequency
of the bandwidth of the channel.
If we work it out,
the math is very simple.
So we have the product, of course,
of the real part and imaginary part of
the baseband signal multiplied by cosine
of omega cn + j sine of omega cn.
And the real part of this equation is
simply the real parts of the baseband
signal multiplied by cosine of
omega c n minus the imaginary part
of the baseband signal
multiplied by sine of omega c n.
Now, you see here,
we're multiplying each component,
the real imaginary part of
the baseband signal, by a carrier and
the two carriers
are orthogonal to each other.
So, we have a cosine carrier and
a sine carrier.
Now, these two carriers are othogonal
because they're shifted by a phase of
90 degrees.
Now, when two things are 90 degrees apart,
they're said to be in quadrature.
And so the first component of the signal
will be called the in phase part and
the second component, the one modulated
by the sine, will be the quadrature part.
And this is the nomenclature
behind the acronym QIM.
Okay, so
now this signal s of n is clearly real and
our next step will be to show that
if we receive this at the receiver
we will be able to recover b of n,
the complex baseband signal.
But before we do that,
let's look at the modulation
process in the frequency domain.
Because the intuition will help us
understand why we can recover the baseband
signal exactly at the receiver.
So in the next few diagrams,
we will show the spectra of br and bi,
the in-phase and quadrature
components of the baseband signal.
Let's assume for the sake of convenience
that these spectra are purely real.
And we will indicate our real
quantities with shades of blue and
purely imaginary quantities
with shades of pink.
The math would stay the same for
arbitrary spectra but this assumption
will allow us to draw a simpler picture.
So if we start by plotting the spectrum
of the real part of the baseband signal,
let's assume it has a shape like this.
Then we plot the spectrum of the imaginary
part of the baseband signal.
And let's assume that, again,
it's a purely real spectrum except that
it has a slightly different shape.
And these shapes
are completely immaterial,
they just help us see what happens
during the modulation process.
The real part is multiplied
by a cosine of omega cn and
therefore a cosine modulation takes place.
It is shifted left or
right and centered in omega c.
For the imaginary part
of the baseband signal,
the modulation takes place with a sine,
and there is a change of sign involved.
So the resulting spectrum will be purely
imaginary, and it will be shifted at omega
c and minus omega c, with a change of sign
in the negative part of the spectrum.
And so this is a spectrum of
the signal that we actually send over
the real channel.
As you can see the signal is real and
indeed
the spectrum has Hermitian symmetry in the
sense that its real part is symmetric and
the imaginary part is anti symmetric.
Okay so now let's assume that
the transmission goes well and
our job at the receiver is to
recover the complex baseband signal.
In either approach we try the usual
method which is multiplying the signal
by the carrier.
Well in this case we have two carriers,
the cosine and sine.
So let's start by
multiplying by the cosine.
And so if we take s[n] which is the
transmitted and then received signal, and
we multiply it by cosine of omega c n.
What we obtain is b r,
the real part of the baseband signal,
multiplied by cosine square of omega
c n minus b i, the imaginary part of
the baseband signal, multiplied by sine
of omega c n times cosine of omega c n.
Now we use some really basic
trigonometry to express these functions
as functions of 2 times omega c n.
So cosine squared of omega c n becomes
1 + cosine of 2 omega c n over 2.
And sine of omega c n cosine of omega
c n becomes sine of 2 omega cn over 2.
And if we rearrange
the terms of this equation,
we obtain one-half times the real
part of the baseband signal,
plus one-half of a term that only
contains carriers at twice omega c n.
Which is really similar to the situation
we had in the simple cosine modulation of
the modulation for a real signal.
Again, it's much more intuitive if we
look at this in the frequency domain.
And so, this was the signal that
was transmitted and then received.
If we multiply this by cosine
of omega c n, we obtain one-half
of the spectrum of the real
part of the baseband signal.
Plus spurious components at
twice the modulation frequency.
Now we know what to do to get
rid of the spurious components,
we just use a low pass filter.
In this case we can use
again a raised cosine.
When we use this we have what is called a
matched filter configuration where we use
the same filter at the receiver
that we used at the transmitter.
This filter will eliminate
the components out of the baseband.
And we will recover the original,
real part of the baseband signal.
We can do exactly the same for
the quadrature component, for
the complex part of the baseband signal.
And in this case, we multiply
the received signal by sine of omega c n.
And if we work out the math with
simple trigonometry once again,
we obtain minus one-half times the
imaginary part of the baseband signal plus
out of band components at twice
the modulation frequency.
Which we can filter out
with a low pass filter.
So, we have proven that we can
recover the baseband signal even when
the original baseband signal is complex.
The final design for
the transmitter is the following.
Bitstream, scrambler, mapper, up sampler.
We multiply the up sample
signal by e to the j omega c n.
We obtain a complex passband signal.
We take the real part, and finally,
we have the sequence of samples
that we can send to the D/A converter,
and then over onto the channel.
The receiver will receive
the analog signal over the channel.
We'll sample it at the proper sampling
rate, and then demodulate it.
So the signal is split into
two parts that are identical.
The first copy will be multiplied
by cosine of omega cn.
And the second copy will be
multiplied by sine of omega cn.
Both will go through a low pass filter
that is really the matched filter
of the upsampling filter
we used at the transmitter.
And this will give us the real part
of the original baseband signal and
the imaginary part of
the original baseband signal.
We multiply this by j and
we sum them together and
now we have the baseband signal,
which will go through a down sampler.
We keep one sample out of K.
And this will give us the estimated symbol
sequence created by the transmitter.
The Slicer will find the closest
alphabet symbol, and
recover the chunk of mbits
associated to the symbol.
And finally, the Descrambler
will undo the randomization and
recover the original user data stream.
So let's see how we can put everything
we've learned so far together and
design a practical system to send
data over the telephone channel.
Suppose that the bandwidth constraint for
the telephone channel
stipulates that we can only
transmit data from 450Hz to 2850Hz.
This gives us a usable bandwidth, W,
of 2,400Hz with a center
frequency Fc of 1,650 hertz.
Now remember, in our all digital paradigm,
we have to pick
a sampling frequency that is at least
twice the highest frequency there.
So, F max * 2.
This would give us a sampling
frequency of at least 5,700 hertz.
But also, we're going to use an upsampling
factor, which is an integer.
And the trick is to pick
a sampling frequency
which is an integer
multiple of the bandwidth.
So if we pick the multiple equal to 3,
then we get the sampling frequency
of 7200Hz, which of course
satisfies Nyquist criterion.
When we translate the specs
into the digital domain,
we find that the modulating
frequency is 0.458 pi.
Now let's tackle the power constraint and
assume that the telephone line
has a maximum SNR of 22dB's.
You have to pick a probability of
error that you can live with and
let's say that we pick 10 to the -6.
If we use QAM, we can use the formula
that we saw in the previous module to
find the size of the alphabet or
alternatively the number of bits
per symbol that we can send.
And the formula is this one, and
when we plug in the values for
the probability of error and
signal to noise ratio,
we find that we can send
at least 4 bits per symbol.
And with 4 bits per symbol we will
have a constellation of 16 points.
So, that would look like this.
The final data rate, remember, is
the baud rate times the bits per symbol.
The baud rate is equal to the bandwidth,
so 2,400, and so
we have a total of 9,600 bits per second.
This is actually an operating mode
of a modem standard called V.24.
It was popular in the 90s, and it is
still sometimes used in fax machines.
Now the question is, are we doing good
with the respect to the maximum amount of
information that we can
send over this channel?
Remember, we used very specific
design choices to derive this
figure of 9,600 bits per second.
A specific modulation scheme, a specific
probability of error and so on, so forth.
What is the best one can do?
Well this is a complex question and
an exhaustive answer would require
several lectures in information theory.
But there is a formula derived by Claude
Shannon in the late 1940s that states
the capacity of a channel given its
bandwidth and its signal to noise ratio.
And this capacity specifies the amount of
information that we can sell reliably,
meaning with an arbitrarily low
probably of error over a channel.
The formula, unfortunately,
is not constructed.
It doesn't tell us how to send this data,
but gives an upper bound on the amount
of information that can
be sent over the channel.
So, for instance, for
the parameters that we used before.
The maximum capacity for that channel
would be 17,500 bits per second.
And with our design scheme,
we're basically hitting half
the capacity of the channel.
The gap can be narrowed if we use
more sophisticated modulation and
data coding techniques.
But as I said, to explore these topics,
we would have to start an entirely
new class in information theory.