Direct-Sequence Spread Spectrum Modulation

Block diagram and equivalent low-pass model

One possibility for realizing a CDMA system is the so-called »Direct-Sequence Spread Spectrum« $\text{(DS–SS)}$ , which is explained here using the block diagram. The corresponding model in the equivalent low-pass range is shown below.

Block diagram and equivalent low-pass model of direct-sequence spread spectrum

In both models,

the distortion-free channel $($AWGN and possibly interference from other users, but no $\text{intersymbol interference}$$)$ is highlighted in yellow,
the $\text{optimal receiver}$ $($matched filter plus threshold decider$)$ is highlighted in green.

Note:

In in the equivalent low-pass model the multiplications with the transmitter-side carrier signal $z(t)$ and the receiver-side carrier signal $z_{\rm E}(t) =2\cdot z(t)$ are omitted.

$\text{This system can be characterized as follows:}$

If the multiplication with the spreading signal $c(t)$ at transmitter and receiver is omitted, the result is a conventional $\text{BPSK system}$ with the carrier $z(t)$ and AWGN noise, characterized by the additive Gaussian noise signal $n(t)$. The second interference component (interference from other participants) is omitted: $i(t) = 0$.
For the following it is assumed $($this is essential for direct-sequence spread spectrum!$)$ that the source signal $q(t)$ has a rectangular NRZ curve. Then the matched filter can be replaced by an integrator over a symbol duration $T$ ⇒ $\text{Integrate & Dump}$. This is followed by the threshold decision.

Principle and properties of band spreading methods

In the following we consider »Pseudo-Noise Band Spreading« in the equivalent low-pass region. So the model outlined below applies.

Low-pass model of direct-sequence spread spectrum

Characteristic for this type of modulation is the multiplication of the bipolar and rectangular digital signal $q(t)$ with a pseudo-random $±1$ spreading sequence $c(t)$:

$$s(t) = q(t) \cdot c(t) \hspace{0.05cm}.$$

The duration $T_c$ of a spreading chip is smaller than the duration $T$ of a source symbol by the integer spreading factor $J$, so that the transmitted signal spectrum

$$S(f) = Q(f) \star C(f)$$

is wider than the spectrum $Q(f)$ by approximately this factor $J$.

$\text{In this context, please note in particular:}$

In previous chapters, a major goal of modulation has always been to be as bandwidth-efficient as possible.
Here, in contrast, we try to spread the signal over as wide a bandwidth as possible.
The bandwidth expansion by $J$ is necessary to allow several subscribers to use the same frequency band simultaneously.
Ideally, $2^J$ suitable spreading sequences can be found. This makes a CDMA system for $2^J$ simultaneous users feasible.

Band spreading techniques also offer the following advantages:

An additional low-rate "DS–SS signal" can be transmitted over a frequency band that is otherwise used by FDMA channels with a higher data rate without significantly disrupting the main applications. The band spread signal virtually disappears under the noise level of these signals.
Targeted narrowband interferers ("sinusoidal interferers") can be combated well with this technique. This military point of view was also decisive for the invention and further development of band spreading techniques.
Furthermore, the band spreading technique in general, but especially $\text{frequency hopping}$ $($fast discrete change of the carrier frequency over a wide range$)$ and $\text{chirp modulation}$ $($continuous change of the carrier frequency during a bit interval$)$ also offer the possibility of better transmission over frequency-selective channels.

Signal curves with a single participant

A disadvantage of direct-sequence spread spectrum modulation is that under unfavorable conditions interference can occur between the subscriber under consideration and other subscribers.

This case is taken into account in the model by the interference quantity $i(t)$.
We initially consider only one transmitter, so that $i(t) = 0$ is to be set.

Signals of direct-sequence spread spectrum modulation in the noise-free case

$\text{Example 1:}$ The graph shows

above the source signal $q(t)$ – marked by blue background – and the band spread signal $s(t)$ as solid black line,
at the bottom left the signal $b(t)$ after band compression,
bottom right the detection signal $d(t)$ after the integrator, directly before the decision.

Further notes:

Discrete-time and normalized signal representation with rectangles spaced by the chip duration $T_c$ is chosen.
The spreading factor is $J = 8$.
As spreading sequence the $\text{Walsh function no. 7}$ is used.
All images show the noise-free case ⇒ $n(t) = 0$.

To the individual signal curves is to be noted:

The $±1$ data signal $q(t)$ is marked by the blue background. After multiplication with the spreading signal $c(t)$, the result is the transmitted signal $s(t)$ which is higher in frequency by the factor $J = 8$.
The spreading signal $c(t)$ is periodic with $T = J · T_c$ and thus has a line spectrum. In the data bits $1,\ 4,\ 8$: $s(t)=c(t)$, other times, $s(t) = - c(t)$.
After band compression at the receiver, i.e., after chipsynchronous multiplication by $c(t) ∈ \{±1\}$ ⇒ $c^2(t) = 1,$ the signal $b(t)$ is obtained.
In the distortion-free and noise-free case:

$$b(t) = r(t) \cdot c(t) = s(t) \cdot c(t) = \big [ q(t) \cdot c(t) \big ] \cdot c(t) = q(t) \hspace{0.05cm}.$$

Integrating $b(t)$ over one bit at a time yields a linearly increasing or linearly decreasing signal $d(t)$. The step curve in the right image is solely due to the discrete-time representation.
At the equidistant detection times the $ν$–th amplitude coefficients $a_ν$ of the source signal $q(t)$ are valid in the distortion-free and noise-free case:

$$ d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T}\hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t = a_\nu \in \{ +1, -1 \}\hspace{0.05cm}.$$

Signals of direct-sequence spread spectrum modulation for $10 · \lg \ (E_{\rm B}/N_0) = 6 \ {\rm dB}$

$\text{Example 2:}$ The two lower graphs change significantly from the first example when AWGN noise is considered.

The AWGN parameter is $10 · \lg \ (E_{\rm B}/N_0) = 6 \ \rm dB$. Then

the band compressed signal $b(t)$ is no longer sectionally constant, and
the detection signal $d(t)$ is no longer linearly increasing or decreasing.

After thresholding the samples $d(νT)$, one nevertheless obtains mostly the sought amplitude coefficients. The vague statement "mostly" is quantifiable by the bit error probability $p_{\rm B}$. As

$$b(t) = \big [ s(t) + n(t) \big ] \cdot c(t) = q(t) + n(t) \cdot c(t)$$

and due to the fact that the statistical properties of white noise $n(t)$ are not changed by the multiplication with the $±1$ signal $c(t)$, the same result is obtained again as for the $\text{conventional BPSK}$ without band spreading/band compression, independent of the spreading degree $J$ :

$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B} }/{N_{\rm 0} } } \hspace{0.05cm} \right ) \hspace{0.05cm}.$$

Additional sinusoidal interferer around the carrier frequency

We continue to assume only one participant. In contrast to the calculation in the last section, however, there are now

in addition to the AWGN noise $n(t)$ also
a narrowband interferer $i(t)$ around the frequency $f_{\rm I}$ with power $P_{\rm I}$ and bandwidth $B_{\rm I}$.

In the limiting case $B_{\rm I} → 0$ the power-spectral density of this "sinusoidal interferer" is:

$${\it \Phi}_{\rm I}(f) = {P_{\rm I}}/{2} \cdot \big[ \delta ( f - f_{\rm I}) + \delta ( f + f_{\rm I}) \big ] \hspace{0.05cm}.$$

In a conventional transmission system without band spreading/band compression, such a narrowband interferer would increase the error probability to an unacceptable extent. In a system with band spreading ⇒ "direct-sequence spread spectrum modulation", the interfering influence is significantly lower, since

band compression acts as band spreading at the receiver with respect to the sinusoidal interferer,
thus its power is distributed over a very wide frequency band $B_c = 1/T_c \gg B$,
the additional interfering power density in the useful frequency band $(±B)$ is rather low and can be taken into account by a slight increase of AWGN noise power density $N_0$.

With $T = J · T_c$ and $B = 1/T$ one obtains:

$$p_{\rm B} \approx {\rm Q} \left( \hspace{-0.05cm} \sqrt { \frac{2 \cdot E_{\rm B}}{N_{\rm 0} +P_{\rm I} \cdot T_c} } \hspace{0.05cm} \right ) = {\rm Q} \left( \hspace{-0.05cm} \sqrt { \frac{2 \cdot E_{\rm B}}{N_{\rm 0} } \cdot \left( \frac{1}{1+ P_{\rm I} \cdot T_c/N_0}\right ) } \hspace{0.05cm} \right )\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{SNR degradation:} \ \frac{1}{\big[1 + P_{\rm I}/(J · N_0 · B)\big]}\hspace{0.05cm}.$$

The larger the spreading factor $J$, the smaller the increase in noise power due to the sinusoidal interferer.

Note: This fact has led to the spreading factor $J$ being often referred to "spreading gain" in the literature, compare for example [ZP85]^[1].

These books are mostly about military applications of the band spreading methods.
Sometimes the "most favorable interferer" is mentioned, namely when the degradation is the largest.
However, we do not want to deal with such applications here.

But the above error probability equation can also be applied approximately when an unspread transmission of higher data rate and a spread spectrum system of lower rate operate in the same frequency band: The interfering influence of the former system with bandwidth $B_{\rm I}$ on the spread spectrum system can be treated approximately as a "narrowband interferer" as long as $B_{\rm I}$ is sufficiently small.

$\text{Conclusions:}$

With AWGN noise (and also many other channels), the bit error probability cannot be reduced by band spreading.
In the best case, band spreading results in the same bit error probability as BPSK (without spreading).
For our purposes, band spreading is a necessary measure to be able to supply several subscribers simultaneously in the same frequency band.
In the following, we will only consider the CDMA aspect and therefore continue to speak of the spreading factor $J$ and not of a "spreading gain".

Exercises for the chapter

Exercise 5.2: Band Spreading and Narrowband Interferer

Exercise 5.2Z: About PN Modulation

References

↑ Ziemer, R.; Peterson, R. L.: Digital Communication and Spread Spectrum Systems. New York: McMillon, 1985.

[1] Ziemer, R.; Peterson, R. L.: Digital Communication and Spread Spectrum Systems. New York: McMillon, 1985.

[1]