# What exactly happens with Spread Spectrum Modulation

With regard to DSSS and CDMA(DS), what does it mean to spread a signal over a spectrum of frequencies in terms of individual frequencies? Is it about sending the same signal redundantly on multiple frequencies, or is the data stream actually divided and sent as multiple discreet signals over various frequencies? I've read explanations claiming either.

Direct-sequence spread spectrum (DSSS) is a technique that is used to generate a modulated signal that occupies more bandwidth than would be implied by its information content alone. A DSSS signal is generated via multiplication of a (typically digitally-modulated) baseband signal by another spreading code waveform. The spreading code waveform is constructed such that it is modulated at a higher rate than the baseband signal; this yields the "spreading" of the resulting signal's spectrum.

To see this, recall the multiplication property of the Fourier transform. Let the baseband signal $x(t)$ be multiplied by a spreading waveform $s(t)$:

$$x(t)s(t)\Longleftrightarrow X(f) * S(f)$$

where the $*$ denotes convolution. If $S(f)$ is broadband when compared to $X(f)$, then the result of convolving $X(f)$ with $S(f)$ is that $X(f)$'s spectral content is "smeared" or "spread" across a wider region of the spectrum. The amount of spreading is dependent upon $S(f)$ and therefore the choice of the spreading sequence $s(t)$. So, for example, you could take a 1 kbps BPSK signal, which would normally have a null-to-null bandwidth of 2 kHz, and spread it to instead have a null-to-null bandwidth of your choice: say, 1 MHz.

To answer your most pointed question, you're not really sending a single signal redundantly on multiple carrier frequencies, nor are you dividing the input signal into pieces that get transmitted at multiple carriers. In lieu of a discrete multi-carrier approach, DSSS instead is more of a continuous smearing of the input signal over a large region of spectrum.

If the receiver is also cognizant of the spreading sequence $s(t)$, then it is capable of despreading its observation of the transmission and therefore recovering the original baseband signal. As long as the spreading sequence $s(t)$ does equal zero for any $t$, then in principle, the spreading process is perfectly reversible. In practice, this isn't exactly the case, as there will be some distortion between what is transmitted and what the receiver sees, but it leads to an important point that is often misunderstood about DSSS:

No benefit in bit error performance, measured in $\frac{E_b}{N_0}$ (the typical metric used for digital modulation performance evaluation) is offered by using direct-sequence spread spectrum. Since the spreading process is reversible, then the principle of reversibility states that there can be no improvement in performance of the optimal receiver for the spread signal when compared to the performance of the optimal receiver for the original, unspread signal.

So if there's no direct bit-error performance benefit, why would you want to do this? There are a few common reasons:

• Multiple access: Code division multiple access schemes work using DSSS techniques. CDMA allows multiple transmitted signals to occupy the same bandwidth over the same time period. Transmitters are associated with separate carefully-selected spreading codes; these are also known by the receiver, which uses them to separate the previously-intermingled-in-time-and-frequency signals. CDMA can be made to seem like the greatest thing since sliced bread, but there are some practical issues (most notably the near-far problem) that limit its usefulness. It is, however, used in some cellular networks around the world, for one example.

• Interference mitigation: Signals with high bandwidth are resistant to narrowband interferers on the AWGN channel. A hand-waving time-domain explanation: a wideband signal in the frequency domain will have a narrow autocorrelation function in the time domain, and vice versa. The optimal receiver will use a correlator (often implemented with a matched filter) matched to the transmitted spread-spectrum waveform, so its output (which ideally will contain copies of the signal's autocorrelation function at symbol-spaced intervals) will contain narrow, sharp peaks that will be easy to identify against the oppositely-characterized background interference.

A hand-waving frequency-domain explanation: the spreading process takes the original baseband signal's power and spreads it across a wide region of the spectrum. In the communication channel, narrowband interference finds its way into the signal that eventually is observed. During the despreading process, the DSSS signal is compressed back to the original baseband signal's bandwidth. However, at the same time, the despreading process actually spreads the narrowband interference across the same bandwidth as the original DSSS signal.

So effectively, the narrowband interferer has been transformed into a wideband interferer across the entire DSSS signal bandwidth, which as you recall, is much larger than the baseband (information-carrying) signal's bandwidth. Therefore, much of the interference power is pushed out of band with respect to the signal of interest, which yields a higher signal-to-interference power ratio (and better performance).

• "Discreetness": No, not discrete, discreet. DSSS is often spoken of as being able to "pull signal out of the noise floor." This refers to the ability of a spread-spectrum receiver to transform a wideband DSSS signal back to the original, much narrower waveform. For a given transmitter power, as the transmitted spread signal bandwidth increases, the average power spectral density over the DSSS bandwidth will decrease, perhaps to the point where it is difficult to observe using spectral analysis at the receiver. This property can be useful for a number of reasons.

One aim might be to decrease the probability of the emitter being detected (and possibly intercepted) by an adversary. Another might be to mitigate interference with other, narrowerband systems that operate in the same region of spectrum. An example of this approach is in ultra wideband communications systems, which have increased in popularity in recent years.

CDMA means that multiple signals are spread and "stacked" on top of the same wide-band channel (each signal takes the entire bandwidth). Each original signal can be obtained because the spreading sequences are orthogonal.

Check out http://en.wikipedia.org/wiki/Code_division_multiple_access#Steps_in_CDMA_Modulation for a good example of how this signal "stacking" works.

• so what does it mean for a signal to "take the entire bandwidth"? an analog or baseband signal naturally occupies multiple frequencies, but a digital signal modulated, let's say with QAM, would only require one frequency. – estolua Jul 9 '12 at 22:59
• One carrier frequency, but with a set bandwidth. With DSSS the carrier moves multiple times within the channel. So if you look at it over time, it will occupy the entire channel, e.g. using peak hold on an analyzer. – Aaron D. Marasco Jul 10 '12 at 1:45
• @estolua: "a digital signal modulated, let's say with QAM, would only require one frequency" - This is false. Digitally modulated signals do not have zero bandwidth, as would be implied by your assertion that they would only "require one frequency." Your confusion doesn't seem to be related to spread spectrum at all, but instead is a more fundamental misunderstanding of signals and systems. – Jason R Jul 10 '12 at 1:49
• @JasonR yes that may very well be the case :) so is there an easy answer why a modulation scheme other than something like FSK would need multiple frequencies? I don't really need to get deep into the subject matter but it seems like this would make a lot of things easier to grasp. – estolua Jul 10 '12 at 8:15
• hmmm, I'm sure there is a better-worded fundamental explanation, but basically finite-time filters must have a non-zero bandwidth (a Dirac delta function, or "single frequency", requires infinite-time/non-causal signal which is not practical). – wrapperapps Jul 10 '12 at 13:09

Consider an FFT with N bins covering your entire spectrum, and a signal modulated by a constant, say a '0'. Temporarily ignore the splatter caused by the non-constant modulation required to actually send information. AM and PM might take up one bin out of N in the FFT result. An interfering signal in that one bin might take out your signal. FM might take up 1 out of 2 possible bins, using a different bin for communicating a '1'.

Now consider sending your '0', not in 1 bin, but as an N-dimensional vector using possibly all N bins (and with a '1' as a different mutually orthogonal N dimensional vector).

You might call this "spread" because the total transmission energy is spread out in N dimensions instead of all packed into just 1 bin, making your signal less likely to be found. You might also call this spread because you still might be able to decompose the sum of more than one N-dimensional vector into some orthogonal components, thereby digging your signal out from among other senders using the same N bins within the same spectrum as you (but hopefully using an orthogonal encoding).

Or in the case of OFDM, you might be using a large portion of all the possible combinations of orthogonal N-dimensional vectors to send some fraction of N bits of information in one fat chunk, and with a much lower modulation rate than required for the same data rate with per-bit AM, PM or FM (which greatly helps with splatter and multi-path issues).

The above thought experiment is, of course, a gross oversimplification, but might provide some insight into why the use of multiple frequency bins might not be just a redundant repeating of the data, but a useful encoding of the data into more than just one or two frequencies.

• @estolua Even though you have accepted hotpaw2's answer, the accepted answer is in fact not relevant to the question "What exactly happens with spread spectrum?" It is instead an excellent answer to a completely different question! The "grossly oversimplified thought experiment" has little to do with spread spectrum communication as most people view the notion. See JasonR's answer for a much better description of what spread spectrum is all about. – Dilip Sarwate Jul 12 '12 at 19:47