LOOKING AT DATA SIGNALS
We diplay data signals graphically in many ways, ranging from textbook illustrations to test equipment screens. This note helps you integrate
those views and to see how some modulation parameters affect the appearance of the displays.
Serial transmission is the basis of the demonstration. Recall that serial transmission, in the absence of noise, is modeled like this:
The transmitted signal s(t) is a sequence of overlapping pulses p(t - nT), each multiplied by its data value bn, which is drawn from a specific
constellation. The pulse shape p(t) is usually a square root Nyquist pulse; that is, its Fourier transform is the square root of the Fourier transform of
another pulse q(t) which has the Nyquist property of equispaced zero crossings. The receiver's first step in detection is a filter matched to the
transmitted pulse, so the matched filter's impulse response is p*(-t). The MF output y(t) is also a sequence of overlapping pulses, but with pulse shap
q(t), the pulse autocorrelation function of p(t), and the one with the Nyquist property. We'll see that Nyquist property ensures no interference among
successive symbols, provided the MF is sampled at the right times.
The calculations below use the raised cosine as q(t), the Nyquist pulse, and the square root raised cosine as p(t), the signalling pulse. They are
defined in the course notes. You can select the rolloff parameter (i.e., the excess bandwidth) β below:
β := 0.35
the rolloff (0 < β ≤ 1)
and you can select a modulation, too. BPSK is rather dull, but it's there because it's easy. QPSK is better, 16QAM is more interesting.
j BPSK
k
l
m
n
j QPSK
k
l
m
n
i 16QAM
j
k
l
m
n
hidden calculations
First, a quick look at the Fourier transforms of the pulses. Note that P(f) lacks the symmetry that gives Q(f) the Nyquist property; also Q(f) = |P(f)|2.
Transform P(f) of square root raised cos pulse
1
− 0.5
Transform Q(f) of raised cos pulse
1
− 0.5
0.5
0.5
β = 0.35
0.5
1.25
1
0.75 0.5 0.25
0.5
0
0.25 0.5 0.75
1
1.25
1.25
1
normalized frequency fT
0.75 0.5 0.25
0
0.25 0.5 0.75
1
1.25
normalized frequency fT
Now over to the time domain. Textbooks (like this one) show individual pulses with amplitude equal to the data.
4
4
2
2
0
0
2
4
2
0
1
2
3
4
5
6
7
8
4
This shows the real part of the MF output y(t) with Nyquist pulse
shape q(t). A few individual pulses (which in reality you cannot
observe separately) are shown as broken lines, and the solid line
shows the sum y(t) (of them and other pulses), which can be
observed on an oscilloscope in an analog implementation. Note
that the pulses have the Nyquist property of zero crossings at all
symbol times but their centre - so no interference. At integer
times, the sum equals the value of the pulse centred at that time.
β = 0.35
Now look at the transmitted signal s(t) and the matched filter output y(t) over a longer time interval, with both real (red) and imaginary (blue)
components. You might see a graph like this at the output of a simulation (hey - that's what generated the plot!). Or, if you are working at a
bench, designing and debugging a modem, you might set your storage scope to capture traces like these.
Transmitted signal s(t)
5
3
The real and imaginary components
of the transmitted signals do not go
through the M-QAM data values
(i.e., odd integers) at integer symbol
times, because the pulse p(t) is not
itself Nyquist.
1
1
3
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
β = 0.35
MF output y(t)
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3
1
1
3
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The matched filter output y(t)
does go through the M-QAM
data values at integer symbol
times, because q(t) is a Nyquist
pulse, which eliminates
intersymbol interference. Note
that y(t) can go beyond the data
values between integer symbol
times.
On the other hand, if you are working at a bench, you might set up your non-storage scope with the time base synched to the symbol clock, so you
would see multiple traces on the screen, as shown below, with real (red) and imaginary (blue) . These are called "eye diagrams," since for binary
transmission (BPSK or the real and imaginary parts of QPSK), the pattern looks like an open eye. Both pictures look quite tidy for β = 1, but that
value of rolloff causes wider bandwidth, too.
Eye diagram of Tx output s(t)
Eye diagram of MF output y(t)
5
5
3
3
1
1
β = 0.35
1
1
3
3
5
5
As expected, the Tx output is "blurry," resulting from all
the interference from nearby pulses caused by the nonNyquist nature of the pulse p(t).
Sample the MF output at the times (once per symbol) when the signal
goes through an M-QAM data level (odd integers). No interference
from surrounding pulses, so it's easy to make a decision. If you sample
at the wrong time, it becomes a mess of interfering symbols.
Another useful picture is a parametric plot of the signal on the complex plane. You get these by putting the real part of the signal on the
x-axis and the imaginary part on the y-axis. A vector signal analyzer (VSA) does the same thing when it recovers the complex baseband
signal corresponding to an RF signal. And, of course, you can generate pictures like this from simulations. They look quite tidy for β = 1
(but it's at the cost of increased bandwidth).
Complex plot of Tx output s(t)
5
Complex plot of MF output y(t)
5
0
3
β = 0.35
1
0
1
1
0
1
3
5
Im[y(t)]
Im[s(t)]
3
0
3
5
3
1
1
Re[s(t)]
3
5
5
5
3
1
1
3
5
Re[y(t)]
Try different rolloff values β. For small β, the swings outside
See how the MF output passes through the M-QAM data
the M-QAM grid can be large, forcing the electronics,
values (odd integer + j another odd integer). It also swings
including power amplifiers, to be designed for large dynamic
quite far outside the data grid (2x2 for QPSK, 4x4 for 16QAM)
range.
between those times, especially with small β values.
====================================================================================================
Now repeat with QPSK and β = 1. The bandwidth increases (higher β) and the graphs become tidier (higher β and QPSK).
First, a quick look at the Fourier transforms of the pulses. Note that P(f) lacks the symmetry that gives Q(f) the Nyquist property; also Q(f) = |P(f)|2.
Transform P(f) of square root raised cos pulse
1
− 0.5
Transform Q(f) of raised cos pulse
1
− 0.5
0.5
0.5
β =1
0.5
1.25
1
0.75 0.5 0.25
0.5
0
0.25 0.5 0.75
1
1.25
1.25
normalized frequency fT
1
0.75 0.5 0.25
0
0.25 0.5 0.75
1
1.25
normalized frequency fT
Now over to the time domain. Textbooks (like this one) show individual pulses with amplitude equal to the data.
4
4
2
2
0
0
2
4
2
0
1
2
3
4
5
6
7
8
This shows the real part of the MF output y(t) with Nyquist pulse
shape q(t). A few individual pulses (which in reality you cannot
observe separately) are shown as broken lines, and the solid line
shows the sum y(t) (of them and other pulses), which can be
observed on an oscilloscope in an analog implementation. Note
that the pulses have the Nyquist property of zero crossings at all
symbol times but their centre - so no interference. At integer
times, the sum equals the value of the pulse centred at that time.
4
β =1
Now look at the transmitted signal s(t) and the matched filter output y(t) over a longer time interval, with both real (red) and imaginary (blue)
components. You might see a graph like this at the output of a simulation (hey - that's what generated the plot!). Or, if you are working at a
bench, designing and debugging a modem, you might set your storage scope to capture traces like these.
Transmitted signal s(t)
5
The real and imaginary components
of the transmitted signals do not go
through the M-QAM data values
(i.e., odd integers) at integer symbol
times, because the pulse p(t) is not
itself Nyquist.
3
1
1
3
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
β = 1
MF output y(t)
5
3
1
1
3
5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
The matched filter output y(t)
does go through the M-QAM
data values at integer symbol
times, because q(t) is a Nyquist
pulse, which eliminates
intersymbol interference. Note
that y(t) can go beyond the data
values between integer symbol
times.
On the other hand, if you are working at a bench, you might set up your non-storage scope with the time base synched to the symbol clock, so you
would see multiple traces on the screen, as shown below, with real (red) and imaginary (blue) . These are called "eye diagrams," since for binary
transmission (BPSK or the real and imaginary parts of QPSK), the pattern looks like an open eye. Both pictures look quite tidy for β = 1, but that
value of rolloff causes wider bandwidth, too.
Eye diagram of Tx output s(t)
Eye diagram of MF output y(t)
5
5
3
3
1
1
β =1
1
1
3
3
5
5
As expected, the Tx output is "blurry," resulting from all
the interference from nearby pulses caused by the nonNyquist nature of the pulse p(t).
Sample the MF output at the times (once per symbol) when the signal
goes through an M-QAM data level (odd integers). No interference
from surrounding pulses, so it's easy to make a decision. If you sample
at the wrong time, it becomes a mess of interfering symbols.
Another useful picture is a parametric plot of the signal on the complex plane. You get these by putting the real part of the signal on the
x-axis and the imaginary part on the y-axis. A vector signal analyzer (VSA) does the same thing when it recovers the complex baseband
signal corresponding to an RF signal. And, of course, you can generate pictures like this from simulations. They look quite tidy for β = 1
(but it's at the cost of increased bandwidth).
Complex plot of Tx output s(t)
5
Complex plot of MF output y(t)
5
0
3
β =1
1
0
1
1
0
1
3
5
Im[y(t)]
Im[s(t)]
3
0
3
5
3
1
1
3
5
Re[s(t)]
Try different rolloff values β. For small β, the swings outside
the M-QAM grid can be large, forcing the electronics,
including power amplifiers, to be designed for large dynamic
range.
5
5
3
1
1
3
5
Re[y(t)]
See how the MF output passes through the M-QAM data
values (odd integer + j another odd integer). It also swings
quite far outside the data grid (2x2 for QPSK, 4x4 for 16QAM)
between those times, especially with small β values.