# Textbook Signals

Yes, the CSP Blog uses the simplest idealized cyclostationary digital signal–rectangular-pulse BPSK–to connect all the different aspects of CSP. But don’t mistake these ‘textbook’ signals for the real world.

What good is having a blog if you can’t offer a rant every once in a while? In this post I talk about what I call textbook signals, which are mathematical models of communication signals that are used by many researchers in statistical signal processing for communications.

We’ve already encountered, and used frequently, the most common textbook signal of all: rectangular-pulse BPSK with independent and identically distributed (IID) bits. We’ve been using this signal to illustrate the cyclostationary signal processing concepts and estimators as they have been introduced. It’s a good choice from the point of view of consistency of all the posts and it is easy to generate and to understand. However, it is not a good choice from the perspective of realism. It is rare to encounter a textbook BPSK signal in the practice of signal processing for communications.

I use the term textbook because the textbook signals can be found in standard textbooks, such as Proakis (The Literature [R44]). Textbook signals stand in opposition to signals used in the world, such as OFDM in LTE, slotted GMSK in GSM, 8PAM VSB with synchronization bits in ATSC-DTV, etc.

Typical communication signals combine a textbook signal with an access mechanism to yield the final physical-layer signal–the signal that is actually transmitted (My Papers [11], [16]). What is important for us, here at the CSP Blog, is that this combination usually results in a signal with radically different cyclostationarity than the textbook component. So it is not enough to understand textbook signals’ cyclostationarity. We must also understand the cyclostationarity of the real-world signal, which may be sufficiently complex to render mathematical modeling and analysis impossible (at least for me). (See also some relevant examples of real-world signals here and here.)

So what, exactly, defines a textbook signal? It is a model for a baseband, IF, or RF signal that has no transmitter impairments, has IID symbols, and has no effects related to a multiple access mechanism. Transmitter impairments (The Literature [R38]) include phase noise, carrier-frequency drift, symbol-clock jitter, and gain/phase mismatch. These effects typically weaken the cyclostationarity of the signal, but do not typically introduce new periods of cyclostationarity (new cycle frequencies). The deviation from IID symbols can arise from the nature of the source message, and from the inclusion of periodically repeated symbols that facilitate receiver operations like synchronization and channel equalization. The deviations from IID symbols can introduce new cycle frequencies relative to the textbook model. Finally, the inclusion of effects related to the access method (frequency-division multiple access [FDMA], time-division multiple access [TDMA], code-division multiple access [CDMA], etc.) can radically add to or change the cycle frequencies and cycle-frequency pattern relative to the textbook signal. A particularly good example is GSM, which combines a Gaussian minimum-shift keyed (GMSK) signal with a TDMA access method.

To illustrate, consider a simulated textbook GMSK signal with bit rate of 250 kHz and carrier frequency 100 kHz (complex-valued data). When the cycle frequencies for this signal are blindly estimated, the following plot is obtained:

The bit-rate non-conjugate cycle frequency is detected by my blind CSP processing, as well as two conjugate cycle frequencies separated by the bit rate of 250 kHz and centered at the doubled carrier (200 kHz), which is consistent with the known cyclostationarity of GMSK/MSK/SQPSK. Here is the outcome of the same blind processing applied to a collected GSM signal:

There are so many cycle frequencies that they can hardly be distinguished from each other. Note that there are two obvious peaks in the conjugate plot, and that their separation is equal to the GSM bit rate of 270.8 kHz. So the underlying GMSK pattern is there, but the overall cyclostationarity for GSM is both quite different and much richer. Zooming in on this plot reveals more structure:

Many of these cycle frequencies are harmonics of the frame rate, which is 216.6 Hz. Some arise from the presence of the GSM midamble, which is one of eight 26-bit sequences that is inserted into the middle of each data slot for channel estimation (equalization) purposes. Some no doubt arise for reasons I don’t understand due to the complex nature of GSM signalling.

Many research papers use textbook signals in their simulation or numerical results sections. (I’ve used them myself when I was assessing the ability of cyclic cumulants to serve as classification features for digital communication signals (My Papers [25] [26] [28])). A recent example is Ramirez et al (The Literature [R45]): “Detection of Multivariate Cyclostationarity”. The authors use a sensor array to detect the presence of a signal with a specific period of cyclostationarity. In the numerical (simulation) results, they show that their derived detector is superior to others taken from the literature. The signal they simulate is a textbook PSK signal: QPSK with rectangular pulses and presumably IID symbols. The reason I cite this particular example is that the textbook nature of the signal is used by the authors as part of the reason for their claims of superiority:

“The signal is … a QPSK signal with rectangular shaping and a symbol rate of … 300 Kbauds.”

“Both LMPIT and GLRT outperform the detectors [28], [37] because they exploit the information at all lags and all harmonics of the cycle frequency. On the contrary, the detector in [37] exploits only the information at one harmonic and one lag.”

This simulated QPSK signal has multiple non-conjugate cycle frequencies that are the harmonics of the symbol rate because it is a textbook rectangular-pulse PSK signal. Rectangular-pulse PSK signals possess, theoretically, an infinite number of cycle frequencies (harmonics of the symbol rate). More realistic, but still textbook, QPSK signals would use a square-root raised cosine pulse-shaping function, which results in a single harmonic of the symbol rate (only one non-conjugate cycle frequency besides the trivial one of zero). So here we have claimed algorithm superiority based at least in part on the textbook signal choice. But … maybe I’m just mistaken about the prevalence of textbook (in particular rectangular-pulse) signals in the world.

I'm a signal processing researcher specializing in cyclostationary signal processing (CSP) for communication signals. I hope to use this blog to help others with their cyclo-projects and to learn more about how CSP is being used and extended worldwide.

## 5 thoughts on “Textbook Signals”

1. Mirko von Leipzig says:

I can think of no example where a rectangular pulse is used in practice. Perhaps in purely digital signals? But I would think there it would be purely BPSK. Clock circuits use rectangular pulses, but they aren’t IID at all (in fact completely the opposite).

Well formed real world signals (in the satellite comms domain) are almost exclusively square root raised-cosine pulse where the symbols are IID (or as close as we can make them) due to the scrambling and encoding used.

There are odd scenarios in which the symbols aren’t IID. An example is the DVB-S2 standard. The header is BPSK, which is not encoded or scrambled (I think – this was a while back). The header is a small portion compared to the actual frame (which is scrambled, encoded) so overall the payload is still mostly IID.

1. Agree on the use of rectangular pulses, but would still like others to chime in …

“Well formed real world signals (in the satellite comms domain) are almost exclusively square root raised-cosine pulse where the symbols are IID (or as close as we can make them) due to the scrambling and encoding used.”

But how is synchronization performed, or channel estimation/equalization, without a periodically transmitted training sequence? Blind techniques? In terrestrial systems, the framing and associated training sequences (preambles, midambles, etc.) are the main culprits that destroy the IID nature of a textbook signal.

I’ll take a look at DVB-S2…

2. Todd Reinking says:

Looks like this thread is pretty stale, but I’ll just throw out a few observations. Yes, the modems supporting the DVB-S standard generate fairly “textbook” signals IN the DSP section of the modem. But once it passes through the RF section of the modem it has all the transmitter impairments listed above (I/Q imbalance is often another important impairment). In a satellite ground terminal, RF impairments will be imprinted on the transmit waveform all through out the RF chain, beyond the modem itself.

“But how is synchronization performed, or channel estimation/equalization, without a periodically transmitted training sequence? Blind techniques? In terrestrial systems, the framing and associated training sequences (preambles, midambles, etc.) are the main culprits that destroy the IID nature of a textbook signal.”

DVB-S2 includes a frame-header (SOF and PLSCODE) for frame synchronization. It also allows the option for inserting pilot symbols. However, as Mirko noted, the frame-header is pretty short. So, at least for OTA signals, I would think that SATCOM signals, such as DVB-S2, are going to be about is “textbook” as you’ll find.

When thinking about CSP, it’s worth noting that modems supporting DVB-SX (follow on to DVB-S2) include pulse-shaping with excess-bandwidth as low as 5% (not sure how much this is used in practice), limiting spectral correlation as noted in the “Cyclostationarity of Digital QAM and PSK | Cyclostationary Signal Processing” article on this site.

BTW, great site, Chad. I’m really enjoying the material I’ve read so far.

1. Thanks for the comment, Todd, and for visiting the CSP Blog. The thread is no longer stale!

Regarding DVB-S2, is it easy for you to specify both the length of the frame-header and the length of the frame?
Or, even better, can you point us to a site that has captured examples of the signal?

Yes, the 5% EBW is quite small, which would push us, perhaps, to performing the initial signal detection and cycle-frequency estimation using fourth-order statistics. Even signals with 0% EBW can still exhibit cyclic features at higher orders.

1. Todd Reinking says:

“Regarding DVB-S2, is it easy for you to specify both the length of the frame-header and the length of the frame?”

DVB-S2 supports a “normal” and “short” payload size. I don’t recall there being a way to adjust the header length. The header contains two things: a fixed start-of-frame sequence and information about modulation and coding for the payload to support adaptive coding and modulation. The DVB-S2 specification is freely available: https://tinyurl.com/y9b2eqgy or Duckduckgo ETSI TR 102 376-2.

“Or, even better, can you point us to a site that has captured examples of the signal?”

Hmm, I’m not aware of such a site. There is a DVB-S2 transmitter module for GNU Radio, however. I’ve not used it, so I can’t vouch for the correctness of it.

“Even signals with 0% EBW can still exhibit cyclic features at higher orders.”

I didn’t know that. It pays to post!