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 b n, 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 :=. the rolloff ( < β ) and you can select a modulation, too. BPSK is rather dull, but it's there because it's easy. QPSK is better, 6QAM is more interesting. nmlkjbpsk nmlkjqpsk nmlkji6qam
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). Transform P(f) of square root raised cos pulse.. Transform Q(f) of raised cos pulse.... β =...7.....7. normalized frequency ft..7.....7. normalized frequency ft Now over to the time domain. Textbooks (like this one) show individual pulses with amplitude equal to the data. 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. 6 7 8 β =.
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) 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. 6 7 8 9 6 β =. MF output y(t) 6 7 8 9 6 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 =, but that value of rolloff causes wider bandwidth, too. Eye diagram of Tx output s(t) Eye diagram of MF output y(t) β =. As expected, the Tx output is "blurry," resulting from all the interference from nearby pulses caused by the non- Nyquist 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 = (but it's at the cost of increased bandwidth). Complex plot of Tx output s(t) Complex plot of MF output y(t) Im[s(t)] β =. Im[y(t)] Re[s(t)] 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 (x for QPSK, x for 6QAM) range. between those times, especially with small β values. ====================================================================================================
Now repeat with QPSK and β =. 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). Transform P(f) of square root raised cos pulse.. Transform Q(f) of raised cos pulse.... β =..7.....7. normalized frequency ft..7.....7. normalized frequency ft Now over to the time domain. Textbooks (like this one) show individual pulses with amplitude equal to the data. 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. β =
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) 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. 6 7 8 9 6 β = MF output y(t) 6 7 8 9 6 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 =, but that value of rolloff causes wider bandwidth, too. Eye diagram of Tx output s(t) Eye diagram of MF output y(t) β = As expected, the Tx output is "blurry," resulting from all the interference from nearby pulses caused by the non- Nyquist 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 = (but it's at the cost of increased bandwidth). Complex plot of Tx output s(t) Complex plot of MF output y(t) Im[s(t)] β = Im[y(t)] Re[s(t)] Re[y(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. 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 (x for QPSK, x for 6QAM) between those times, especially with small β values.