# Use of pulse shaping in digital modulation

Sorry, I am a little bit confused here cause I thought when we say QPSK, the signal looks like the following. I thought the bits to send/receive depends on the phase shift of the carrier, or is the above picture wrong or will never happen in real world?

So the discussion from here is to use pulses to send 1's and 0's? What does the signal look like in air actually? Pulse Shaped or like my attached picture? I am confused, so does the phase shifted carrier signal indicate a symbol or does the amplitude of a pulse shaped signal carries the bits?

QPSK in its raw form would appear just as the OP has shown, where the information is encoded into four symbols given by the four phase states alone. The problem is this has a very wide transmit spectrum given the pulse shapes are rectangular and the Fourier Transform of a rectangular pulse is a Sinc function (whose peaks roll-off in frequency only at the rate of 1/f). Wireless spectrum is an expensive commodity, so we go through great lengths in terrestrial wireless communications to limit spectral occupancy and in the process improve spectral efficiency. This is the only reason for pulse shaping, but it is significant. By more slowly transitioning between symbols rather than the abrupt transitions shown, we greatly limit the amount of spectrum needed (the spectrum will be given by the Fourier Transform of the pulse shape). Effective pulse shaping does this while controlling the inter-symbol interference that is added in the process.

The graphic below demonstrates this for a 16QAM waveform where the real portion of the time domain waveform is shown in the upper part of the graphic with and without pulse shaping. The lower part of the graphic shows comparative spectrum where we see the significant reduction in spectral occupancy due to the pulse shaping provided. I see this graphic is the same one as what the OP linked, so to further explain: This is how the envelope of the signal would look like in the air (we don't see the actual carrier frequency in this graphic). This graphic was for QAM so does have information in the amplitude as well as phase, but for QPSK case the amplitude would carry no information. But even with the QAM case, the pulse shaping modifies the amplitude ONLY to reduce the spectrum, and modifies it in such a way to control the trajectory between our symbol samples of interest, while still passing through those samples defined by the modulation exactly at the proper symbol timing locations.

Below is an eye diagram showing the real and complex waveform for the QPSK case (The eye-diagram similar to the time domain waveform shown above, but repeating synchronous to symbol boundaries) Show this shows how the over the air waveform would look (in actuality we typically do half the pulse shaping in the transmitter with a root-raised cosine filter while this plot shows the result after the second root-raised cosine filter in the receiver), but in general we are seeing the amplitude and phase of the time domain waveform for all samples, and the job of the receiver is to determine the sampling locations where the red dots are shown (the center of each symbol), and we then determine the magnitude and phase of those to demodulate (or demap) the QPSK data.

To see what is transmitted "over the air", the sketch below begins to demonstrate how the envelope of the carrier would be modulated by the pulse shaping, but doesn't provide the whole picture as the phase would also be transitioning from symbol to symbol so would not align with what would occur if we continued to transmit one symbol repeatedly. I added more details further below showing the exact waveforms expected. These are the real waveforms that are transmitted at the antenna rather than the baseband equivalent waveforms showing the complex envelope typically with two plots as I (real) and Q (imaginary). The sinusoidal carrier could be any frequency and the envelope would not change, including frequency = 0 which would then represent the eye diagram and I/Q plots that I have shown above (and must then be a complex waveform to represent). A further note is the OP used symbol mapping that is not optimally gray coded (the 180 degree transition depicted in the diagram going from symbol 00 to 10 should be reserved for transitions where both bits change such that only one bit changes for all symbols that are close to each other).

Also note how this is similar to the operation of windowing prior to computing the FFT to reduce spectral leakage.

To avoid any confusion based on subsequent questions in the comments, here is the actual results on what the OP's QPSK waveform would look like at "RF" using properly gray-coded symbols using the following mapping of symbol to constellation:

0: -1-1j

1: 1-1j

2: -1+1j

3: 1+1j

We will start with the final waveform in the receiver as the desired QPSK constellation and work back to what is transmitted over the air by adding in the carrier offset and seeing the result:

First the I and Q waveform at baseband for a QPSK waveform pulse-shaped with a raised cosine filter with alpha = 0.3 would appear as the following in the receiver once all the carrier offsets are removed: The constellation of the above is formed by plotting I vs Q on the complex plane. These trajectories represent the phase and amplitude of our RF carrier at every moment in time, and here I show the one instance that occurs at every integer symbol number in red, which is the only point over the duration of each symbol where the phase and magnitudes of the carrier will actually represent what we expect for QPSK (every other sample in between will be in transition at other phases and amplitudes!). With an actual carrier frequency added, this same sequence would become the waveform shown in the plot below, where the amplitude of the envelope will match the amplitude of the constellation in the plot above: Zooming in on the first 10 samples, this is what the OP would see for the sequence given (this is for the sequence 0,2,1,3,0,2,1,3,1,1,2,2.... specifically). The symbol sample points are aligned in this plot with the integer symbol locations on the horizontal axis. This was done with a carrier frequency that is 10 times the symbol rate, but if we increased the carrier frequency further this plot showing the envelope would just get solid but appear identically the same otherwise. And the same thing as we reduce the carrier to 4 times the symbol rate, showing that we will see the same basic envelope shape regardless of which carrier frequency is used (and as noted if the carrier gets too small compared to the symbol rate we need to use a complex signal to properly represent the waveform, exactly as we do in transceiver implementations): Note that these are all done with a raised cosine filtered pulse-shape, but what is typically transmitted over the air is root-raised cosine filtered. To be complete, below is the plot showing the same waveform with a root-raised cosine pulse shape: With close observation of the above plot against the unmodulated carrier aligned with the first symbol we see how the modulated carrier is continuously changing in phase during the course of each symbol and, as also made clear by the constellation diagram of the modulated waveform, is only at the correct amplitude and phase according to the QPSK constellation only over very short durations, and in this specific case after being passed through a second RRC filter. (If the pulse-shape is altogether eliminated then the waveform will stay at the correct amplitude and phase for the entire symbol, and as the pulse shape roll-off factor is increased, this will approach that). The above plots and descriptions used Raised Cosine (and Root-Raised Cosine) pulse shaping but an alternate harris-Moerder filter is described in this paper http://eon.sdsu.edu/~seshagir/SDR05.pdf. Using this as the pulse shaping filter results in an order of magnitude lower EVM for the same number of taps. This isn't commonly used even today (15 years later) to my knowledge since the root-raised cosine filter is baked into our specifications but certainly is very compelling to be aware of for consideration in future radios (I added a question to see if there is further experience with its use: Use of the harris-Moerder Nyquist Pulse Shaping Filter which may have additional information on this.)

• The blue piecewise constant graph under a Pulse Shaping header resembles a 4-PAM signal waveform, if anything, and certainly does not represent a sample of 16QAM waveform. Neither in its "raw form", nor any way processed. Sep 19 '20 at 3:23
• @V.V.T of course it does- as I stated this is the real part of the waveform which is 4-PAM for a 16QAM complex waveform (4x4 = 16) Sep 19 '20 at 7:54
• (Just like the eye diagram below shows a 2 level waveform on I and Q for QPSK... for the 16-QAM version I only show I (or Q) Sep 19 '20 at 7:55
• @DanBoschen Thanks for the explanation and the effort you put in it! It really helped alot!! Sep 21 '20 at 10:07
• @увевонг The other answers are also helpful and I think VVT's response will make more sense to you if he used a lower roll-off factor for the bottom plots in each figure (as then they would have envelopes that appear as in my and user67081's answer, tying it all together). You can have ANY number of cycles within the envelope (even fractional) and it would still be QPSK as we would observe it in the transceiver. For small cycles (representing a small carrier offset) it must be represented as a complex signal. Sep 21 '20 at 14:07

If you actually were to scope the output of say a RF software defined radio transmitting pulse shaped BPSK (similar for QPSK) - you'll see something like the top trace "RF Out" below: A couple notes to explain: you can't see it unless zoomed in but the rf carrier is oscillating very fast which is why it appears solid, however it's envelope traces out the shape as shown. Note that if we were to turn off pulse shaping, the signal would look exactly like the one you posted initially.

I like to think of pulse shaping as taking your initial signal and squeezing down the amplitude to near zero at the points where the phase changes. This is useful because instead of having a very sharp transition in the time domain (which results in a very wide bandwidth spectral content), your phase transitions are much more smooth. This is why we get the big reduction in the sidelobes in the frequency domain as shown in Dan Boschen's first plot. Another way to think about this is that unfiltered qpsk is using a rect pulse (sinc in frequency, with sidelobes extending out) whereas we might instead use the root raised cosine pulse which does not exhibit the sidelobes in the frequency domain and takes up less spectral content.

Edit: I should also note since you mentioned a confusion about amplitude vs phase carrying the info. BPSK can also equivalently be considered as binary amplitude modulation. You can see this pretty clearly from looking at the 4th trace, you could easily demodulate this signal by thresholding between <0 vs >0 (of course you may have 180 degree ambiguity but that's common and has to be corrected anyway either via preamble or differential encoding). With respect to QPSK, it's slightly different since it would be thought of as the sum of in-phase and quadrature BPSK signals. So in other words QPSK could be considered amplitude modulation on a quadrature carrier.

• Thanks for the explanation! It's very helpful! Sep 21 '20 at 10:06

The QPSK signal never looks like your attached picture. Neither in the transceivers, nor "in the air".

First, I cannot remember an application where the QPSK symbol rate is on the order of the radio frequency. The 802.11a's symbol duration is 4 usec, with 5GHz carrier it gives 20,000 wave periods. The bandwidth of conventional narrowband and carrier wave transmissions is usually a tiny fraction of the carrier bandwidth. Otherwise, this communication is called Ultra Wide Band, a rapidly advanced technology.

This is not to say that in simulation we cannot modulate a carrier with the signal at rates that high. It may have no practical use, bit still be an instructive experience, to perform these simulations "by hand" and visually inspect the results. The two pictures below show the results of this simulation. The first picture follows your attached picture in that there are two carrier wave periods per symbol.

• The top graph is the (real part of) product of complex symbol amplitude and the carrier. Notice that the phases in QPSK are 45°, 135°, 225°, and 315°, in contrast to 0°, 90°, 180°, and 270° of those shown in your picture. I will not elaborate on the subject, only add that there are no pure carrier wave present in the transmission that one would be able to filter out from the antenna signal. The "carrier waves" at the centroid lines of the signal graphs can be considered "ghost carriers".
• The middle graph is the symbol-carrier product filtered out with the sinc filter (brick filter in the frequency domain).
• The bottom graph is the symbol-carrier product filtered out with the pure raised cosine filter, the roll-off factor = 1. The second picture shows the waveforms with eight carrier wave periods per symbol, the intermediate case between a narrowband transmission and UWB.

The answers would be more more helpful and relevant, if you first try to run simulations for yourself, and ask the SE community for help in writing the simulation code, if you need, or for interpreting simulation results. Instead, with the question like this, the posters often guess what you really need to know.