6.1. TX Pulse Shaping#
6.1.1. Power spectral density of digitally modulated signal#
To see how we may control the spectral shape of the TX signal, we need to first obtain the power spectral density (PSD) of a digitally modulated signal. We will work in the complex baseband here throughout.
Let \(b[n]\) be a stationary sequence of random data symbols that we wish to transmit. Let \(T\) be the symbol period. For a symbol period, we transmit one signal from the finite collection \(\{s_0(t), s_1(t), \ldots, s_{M-1}(t)\}\) to convey one symbol worth of data. We usually call the vector-space representation of the collection of signals, or just the collection itself, the signal constellation. The choice of which signal to send is determined by the whole symbol sequence \(b[n]\) in general. For the \(n\)th symbol period, denote the signal transmitted by \(s(t-nT;n)\), where the delay \(nT\) accounts for transmission time of this signal and the second argument \(n\) should be interpreted as the signal choice from the constellation as discussed for this \(n\)th symbol period.
The TX signal is then
(6.1)#\[\begin{equation} x(t) = \sum_{n = -\infty}^{\infty} s(t-nT; n). \end{equation}\]The autocorrelation function of \(x(t)\) is \(R_{x}(t+\tau,t) \triangleq E[x(t+\tau)x^*(t)]\), which is clearly periodic (with period \(T\)) in \(t\). Hence, \(x(t)\) is a wide-sense cyclostationary process.
Consider an observer located at some random location from the TX (not too far away so that the path loss is minimal). The TX signal observed by this observer can then be modeled as \(x(t-T_d)\), where \(T_d\) is a random delay uniformly distributed over the interval \([0,T]\), independent of \(x[n]\). It can be readily shown (see [1] Section 3.4 and [2] Section 2.3) \(x(t-T_d)\) is a wide-sense stationary process whose autocorrelation function
(6.2)#\[\begin{equation} \label{} \bar R_x(\tau) = \frac{1}{T} \int_0^T R_x(t+\tau,t) dt = \frac{1}{T} \sum_{n=-\infty}^{\infty} g_k(\tau -nT), \end{equation}\]where
(6.3)#\[\begin{equation} g_k(\tau) = \int_{-\infty}^{\infty} E[ s(t+\tau;k) s^*(t;0) ] dt. \end{equation}\]The PSD of \(x(t)\) will be defined as the PSD of \(x(t-T_d)\), both denoted by \(S_x(f)\) which is the Fourier transform (FT) of \(\bar R_x(\tau)\). This definition of PSD for the TX signal \(x(t)\) makes physical sense since its spectrum is measured by some instrument located at some distance away from the TX.
Let the FT of \(g_k(\tau)\) be \(G_k(f)\). Then
(6.4)#\[\begin{equation} S_x(f) = \frac{1}{T} \sum_{k=-\infty}^{\infty} G_k(f) e^{-j2\pi k T}. \end{equation}\]Note that the same development can be redone by deterministic time-averaging with the PSD of \(x(t)\) defined as \(\lim_{D\rightarrow \infty} \frac{\left| X_D(f) \right|^2}{D}\), where \(X_D(f)\) is the FT of \(x_D(t) \triangleq \begin{cases} x(t) & \text{if } -\frac{D}{2} \leq t \leq \frac{D}{2} \\ 0 & \text{otherwise} \end{cases}\). The PSD defined this way will still have the same expression in (6.4) (see [2] Section 2.3 for details). This deterministic definition provides us a measurement-based physical meaning for the PSD of \(x(t)\).
6.1.2. Linear modulation and Pulse shaping#
For a linear modulation, \(s(t;n) = b[n] p(t)\), where \(p(t)\) is called the TX pulse shape. Thus, the set of possible symbol values is the constellation. From (6.3),
\[\begin{equation*} g_k(\tau) = R_b[k] \int_{-\infty}^{\infty} p(t+\tau)p^*(t)dt \end{equation*}\]and \(G_k(f) = R_b[k] \left|P(f)\right|^2\), where \(R_b[k] \triangleq E[b[k]b^*[0]]\) and \(P(f)\) is the FT of the TX pulse shape \(p(t)\). Furthermore, using (6.4), we have
(6.5)#\[\begin{equation} S_x(f) = \left|P(f)\right|^2 \frac{S_b \left( e^{j2\pi fT} \right)}{T} \end{equation} \]where
\[\begin{equation*} S_b\left( e^{j2 \pi \hat f} \right) = \sum_{k=-\infty}^{\infty} R_b[k] e^{-j2\pi k \hat f} \end{equation*}\]is the discrete-time Fourier transform (DTFT) of \(R_b[k]\). The quantity \(\frac{S_b \left( e^{j2\pi fT} \right)}{T}\) can be interpreted as the PSD of the data symbol sequence \(b[n]\).
For BPSK, \(b[n]\) are i.i.d. symbols selected from \(\{-1,+1\}\) with equal probabilities. For DBPSK, \(b[n] = a[n] b[n-1]\), where \(a[n]\) are i.i.d. symbols selected from \(\{-1,+1\}\) with equal probabilities. For these two binary (linear) modulations, \(R_b[k] = \delta[n]\) and \(S_b\left( e^{j2 \pi \hat f} \right) = 1\). Therefore, the TX PSD is
\[\begin{equation*} S_x(f) = \frac{1}{T} \left|P(f)\right|^2. \end{equation*}\]The same PSD formula, within a scaling factor, applies to MPSK, DMPSK, and MQAM modulations as well.
For OOK (2-ASK), \(b[n]\) are i.i.d. symbols selected from \(\{0,1\}\) with equal probabilities. So \(R_b[k] = \frac{1}{4} + \frac{1}{4} \delta[n]\) and
\[\begin{equation*} S_b\left( e^{j2 \pi \hat f} \right) = \frac{1}{4} + \frac{1}{4} \sum_{k=-\infty}^{\infty} \delta( \hat f - k). \end{equation*}\]Therefore, the TX PSD is
\[\begin{equation*} S_x(f) = \frac{1}{4T} \left|P(f)\right|^2 + \frac{1}{4T} \left|P(f)\right|^2 \sum_{k=-\infty}^{\infty} \delta \left( f - \frac{k}{T} \right). \end{equation*}\]Note that the PSD of OOK contains spectral lines.
For the calculation of the PSDs of more complicated linear modulations, see [1] Section 3.4.
In general, we see from the simple examples above or from (6.5) that the FT of the TX pulse shape \(p(t)\) controls the spectral shape of any linearly modulated TX signal.
It is also easy to see that for any linear modulation, the TX signal equation in (6.1) simplifies to
\[\begin{align*} x(t) &= \sum_{n=-\infty}^{\infty} b[n] p(t-nT) \\ & = \left[ \sum_{n=-\infty}^{\infty} b[n] \delta(t-nT) \right] * p(t). \end{align*}\]Hence, we can generate the TX signal \(x(t)\) by first modulating an impulse train of period \(T\) with the symbol sequence \(b[n]\) and then passing the symbol-modulated impulse train through a filter whose impulse response is the pulse shape \(p(t)\). This method of generating the linearly modulated signal is called pulse shaping.
6.1.3. Root raised cosine pulse#
A popular choice of pulse shape that allows us to easily tune the spectral shape is the root raised cosine (RRC) pulse defined by the expression below:
(6.6)#\[\begin{equation} p_{RRC}(t) = \frac{\sin \frac{\pi (1-\beta) t}{T} + \frac{4\beta t}{T} \cos \frac{\pi (1+\beta) t}{T}}{ \frac{\pi t}{T} \left[ 1 - \left(\frac{4\beta t}{T} \right)^2 \right] } \end{equation}\]where the parameter \(0 \leq \beta \leq 1\) is called the excess bandwidth or roll-off factor. The FT of the RRC pulse is the RRC spectrum given by
(6.7)#\[\begin{split}\begin{equation} P_{RRC}(f) = \begin{cases} \sqrt{T} & \text{for } |f| \leq \frac{1-\beta}{2T} \\ \sqrt{\frac{T}{2} \left\{ 1 - \sin \left[\frac{\pi T}{\beta} \left( |f| - \frac{1}{2T} \right) \right] \right\} } & \text{for } \frac{1-\beta}{2T} < |f| \leq \frac{1+\beta}{2T} \\ 0 & \text{for } |f| > \frac{1+\beta}{2T}. \end{cases} \end{equation}\end{split}\]We see from (6.7) that the FT of the RRC pulse has a passband from \(-\frac{1+\beta}{2T}\) to \(\frac{1+\beta}{2T}\) Hz. The band edge becomes steeper as \(\beta\) gets closer to \(0\), at which the FT becomes that of an ideal LPF. The support of the RRC pulse is infinite, but its signal magnitude has a \(\frac{1}{t^2}\) decay (except when \(\beta=0\)). In general, the closer \(\beta\) is to \(0\), the longer is the essential support of the RRC pulse.
6.1.4. USRP implementation#
Typically, pulse shaping is implemented in discrete-time in the host computer using a multi-rate filter. The filter output samples are then sent to the USRP for transmission.
The input to the multi-rate filter should be the symbol sequence \(b[n]\) at the symbol rate \(\frac{1}{T}\), while the output contains the signal samples to be sent to the USRP at the TX sampling rate.
The impulse response of the multi-rate filter (in the upsampled domain) is \(h[n] = p\left( \frac{nT}{U} \right)\), where \(U\) is the interpolation factor of the multi-rate filter.
We will use the RRC pulse \(p_{RRC}(t)\) to perform pulse shaping. Because the RRC pulse has an infinite support, we have to approximate the ideal RRC pulse-shaping filter by using a truncated RRC pulse as the impulse response \(h[n]\).
In general, the closer is the excess bandwidth \(\beta\) to \(0\) a longer \(h[n]\) is needed to accurately approximate the ideal RRC pulse-shaping filter. In addition, the expression in (6.6) for \(p_{RRC}(t)\) gives rise to a non-causal filter. Hence we need to delay \(p_{RRC}(t)\) and then sample to obtain a causal \(h[n]\). The usual choice is to delay \(p_{RRC}(t)\) such that the peak of the RRC pulse (at \(t=0\)) appears at the middle sample of \(h[n]\). Remember this delay in the impulse response will introduce a delay in the generated \(x(t)\) by the same amount of time.
Let \(f_s\) be the sampling rate of the USRP. Since the passband of the RRC pulse is from \(-\frac{1+\beta}{2T}\) to \(\frac{1+\beta}{2T}\) Hz, the sampling theorem tells us that we must have \(f_s \geq \frac{1+\beta}{T}\) or equivalently \(\frac{1}{T} \leq \frac{f_s}{1+\beta}\) if we are to perfectly construct \(x(t)\) from the output samples of the multi-rate filter. Thus we would want to set \(\frac{1}{T} = \frac{f_s}{1+\beta}\) and choose \(\beta\) as close to \(0\) as possible so that we can maximize the symbol rate. However, recall that \(h[n]\) needs to be long when \(\beta\) is close to \(0\). Moreover, the upsampling and downsampling factors \(U\) and \(D\) are related to \(f_s\) and the symbol rate \(\frac{1}{T}\) as \(\frac{U}{D} = f_s T\). Hence, we need to set \(\frac{U}{D} = 1 + \beta\). This implies that we will need to use larger integers for \(U\) and \(D\) to make \(\beta\) close to \(0\). Such large choices of \(U\) and \(D\) will undesirably increase the complexity of the multi-rate filter.
Practically we would choose small integers \(U > D\) such that \(\frac{U}{D} -1\) is relatively small. Then we choose \(\beta = \frac{U}{D} -1\) and \(\frac{1}{T} = \frac{D}{U} f_s\). For examples:
Choosing \(U=4\) and \(D=3\) gives \(\beta = \frac{1}{3}\) and achieves a symbol rate of \(\frac{3}{4} f_s\).
Choosing \(U=5\) and \(D=4\) gives \(\beta = \frac{1}{4}\) and achieves a symbol rate of \(\frac{4}{5} f_s\).
Choosing \(U=8\) and \(D=7\) gives \(\beta = \frac{1}{7}\) and achieves a symbol rate of \(\frac{7}{8} f_s\).
Below is a simple C++ function to generate the truncated RRC impulse response based on the design above:
/* Generate a unit-energy truncated RRC impulse response for pulse shaping Assume D <= U <= 2*D This RRC pulse is sampled at U*symbol_rate. Excess bandwidth = U/D - 1 sampling rate / symbol rate = U/D Length of impulse response = 2*len+1 */ void rrc_pulse(std::complex<float>* h, int len, int U, int D) { float beta = float(U - D)/D; //roffoff factor h[len] = 1.0-beta+4.0*beta/M_PI; float scale = std::norm(h[len]); for (int n=1; n<=len; n++) { if (n == U/beta/4.0) { h[len+n] = beta/sqrt(2.0)*((1.0+2.0/M_PI)*sin(M_PI/4.0/beta)+(1.0-2.0/M_PI)*cos(M_PI/4.0/beta)); } else { h[len+n] = (sin(n*M_PI*(1.0-beta)/U) + 4.0*n*beta/U*cos(n*M_PI*(1.0+beta)/U))*U/n/M_PI/(1.0-16.0*n*n*beta*beta/U/U); } h[len-n] = h[len+n]; scale += 2.0*std::norm(h[len+n]); } scale = sqrt(scale); for (int n=0; n<2*len+1; n++) { h[n] /= scale; } }
Remember the TX amplifier on the daughterboard of the USRP is non-linear in the high-gain region. Nonlinearity of the amplifier will introduce spurious emission outside the intended band. Therefore, the TX amplifier gain should not be set to too high if linear operation is required. By default the range of \([-1.0, 1.0]\) of the signal samples corresponds to the full dynamic range of the DAC on the motherboard of the USRP. Thus, linearity can be further assured by reducing the maximum amplitude of the signal samples sent to the USRP.