Lecture 24 - Baseband-Equivalent Models for Lab Kit
1 System Model
Our entire system begins with some discrete input symbols entering a pulse shaping filter. That waveform is then upconverted so it can be trasmitted using an antenna. After propagating through the air (or some other medium), that waveform is received at another antenna and then downconverted to baseband. That baseband signal is then processed to produce a series of discrete symbols.
2 Transmitter Model
We implement pulse shaping using a discrete time approach. Our input symbols are first upsampled by some factor before pulse shaping and then passed into a DAC to produce a continuous time baseband signal.
The model for our baseband signal \(s(t)\) is then
\[ s(t) = \sum_{k=-\infty}^\infty a[k] p(t - kT_s) \]
where \(T_s = M_1 T_1\).
The equivalent discrete time model is
\[ \begin{aligned} s[n] &= \sum_{k=-\infty}^\infty a[k] p(nT_1 - kT_s) \\ &= \sum_{k=-\infty}^\infty a[k] p(nT_1 - k M_1 T_1) \\ \end{aligned} \]
We see that this pulse shape actually looks like a waveform sampled at \(T_1\), so we can rewrite it in discrete time:
\[ s[n] = \sum_{k=-\infty}^\infty a[k] p[n - k M_1] \]
Note that most of the non-ideal effects will be accounted for by modifying the pulse shape model.
We then feed our baseband signal into a PLL and then an I/Q modulator. This passband signal is then fed through an amplifier to increase transmit power. The signal is finally passed through a bandpass filter to reduce the harmonics from the modulator before being transmitted through the antenna.
We end up with a model for the tranmitted signal like the following:
\[ \begin{aligned} x(t) &= G_1 \sqrt 2 \operatorname{Re}\{ s(t) e^{j (2 \pi f_1 t + \varphi_1)} \} \\ &= \frac{G_1}{\sqrt 2} [ s(t) e^{j (2 \pi f_1 t + \varphi_1)} + s^*(t) e^{-j (2 \pi f_1 t + \varphi_1)} ] \\ &= G_1 \sqrt 2 [ s_I(t) \cos(2 \pi f_1 t + \varphi_1) - s_Q(t) \sin(2 \pi f_1 t + \varphi_1) ] \\ \end{aligned} \]
3 Antennas and Propagation
Whenever we have antennas transmitting and receiving in a real environment, their signals are constantly interacting with that environment. The transmitted signal will reflect off of the many different surfaces in the environment before reaching the receive antenna. Because the signal takes multiple paths of different lengths, the receive antenna actually sees multiple copies of the same signal arriving at different times. We call this effect multipath propagation.
summation model of multipath with delayed signals 28:00 We model multipath effects using a summation like the following, where \(y(t)\) is the received signal:
\[ y(t) = \sum_{l=1}^L \alpha_l \, x(t - \tau_l) \]
Where \(L\) is the number of paths on which the signal propagates from transmit to receive, \(\tau_l\) is the delay resulting from a longer travel distance on a given path, and \(\alpha_l\) is the attenuation factor for the path which accounts for all other effects (e.g., antenna gains, polarization loss, propagation loss, etc.).
Although in principle we could compute these properties exactly, oftentimes we treat them as variables and estimate them at the receiver so that we have an acceptable signal. In cases where the receiver is moving (e.g., mobile phones), the properties are constantly changing, so computing them a priori would be nearly impossible.
4 Receiver Model
The receiver begins with a continuous time signal from its antenna. This signal is fed into a bandpass filter to isolate the actual signal. The filtered signal is immediately fed into a low-noise amplifier (LNA) to amplify signal without introducing too much noise. The signal is then demodulated down to baseband and low-pass filtered. After that, we pass the signal into our interface board (which has some gain) which then pushes the signal into the ADC on the ADALM.
The received power of a signal varies based on many physical effects like distance from transmitter, antenna polarization, etc. For this reason, in general, the gain following the LPF in a radio will be variable so that the full dynamic range of the ADC can be used. We would use a variable gain amplifier (VGA) to acheive this.
For simplicity, we will make a two assumptions.
- There is only one propagation path from transmitter to receiver, i.e. \(L = 1\)
- The receiver is linear
Because we have
\[ x(t) = \sqrt 2 \operatorname{Re} \{ s(t) e^{j(2 \pi f_1 t + \varphi_1)} \} \]
we can model the multipath effects like so:
\[ \begin{aligned} \alpha_1 x(t - \tau_1) &= \sqrt 2 \operatorname{Re} \{ \alpha_1 s(t - \tau_1) e^{j(2 \pi f_1 (t - \tau_1) + \varphi_1)} \} \\ &= \sqrt 2 \operatorname{Re} \{ (\alpha_1 e^{-j2 \pi f_1 \tau_1}) s(t - \tau_1) e^{j(2 \pi f_1 t + \varphi_1)} \} \\ \end{aligned} \]
From this equation, we see that gain and delay at passband is gain, phase shift, and delay at baseband.
We can write the expression for \(z(t)\) (the baseband signal before filtering) as follows:
\[ \alpha_1 G_1 G_2 [s(t - \tau_1) e^{-j 2 \pi f_1 \tau_1} e^{j (2 \pi (f_1 - f_2)t + (\varphi_1 - varphi_2))} + s^*(t - \tau_1) e^{j 2 \pi f_1 \tau_1} e^{-j (2 \pi (f_1 + f_2)t + (\varphi_1 + varphi_2))}] \]
Notice that the conjugate term is centered around \(f_1 + f_2\) in the frequency spectrum, so it will be eliminated by the LPF. Therefore, the signal part of \(r(t)\), \(r_s(t0)\) will be
\[ r_s(t) = G_1 G_2 G_3 [ \alpha_1 e^{-j 2 \pi f_1 \tau_1} s(t - \tau_1) e^{j (2 \pi (\Delta f)t + \Delta \varphi)}] \]
Notice the four non-idealities present in this equation:
- Propagation effects (\(\alpha_1 e^{-j 2 \pi f_1 \tau_1}\))
- Delay (\(s(t - \tau_1)\))
- Frequency offset (\(\Delta f\))
- Phase offset (\(\Delta \varphi\))
The signal \(r(t)\) will also have some noise component, which we will call \(r_w(t)\). The entire received signal is then
\[ r(t) = r_s(t) + r_w(t) \]
We model \(r_w(t)\) as AWGN with total noise power \((4{kTRB}) {NF} G_2^2 G_3^2\) (note that gain is squared because this is a power gain). Here, \({NF}\) is the overall cascaded noise figure of the receiver, \(k\) is Boltzmann’s constant, \(T\) is temperature, \(R\) is the resistance of the system, and \(B\) is the bandwidth of the signal.
The last step in the receiver hardware is to sample the waveform so that we have a discrete time signal to process. Recall that our signal is made up of some sequence of symbols that is upsampled, pulse shaped, and then converted to a waveform.
Mathematically, it looks like this:
\[ s(t) = \sum_{k=-\infty}^{\infty} a[k] p(t - kT_s) \]
If we sample the signal at time \(n T_2\), we would get the value \(r[n]\), which is the same as \(r(nT_2)\), which can be written in terms of signal and noise as \(r_s(nT_2) + r_w(nT_2)\)
The signal component is then
\[ r_s[n] = G_1 G_2 G_3 \left[\sum_{l=1}^{L} \alpha_l e^{-j 2 \pi f_1 \tau_l} \sum_{k=-\infty}^{\infty} a[k] p(nT_2 - \tau_l - kT_s) \right] e^{j(2 \pi (\Delta f)nT_2 + \Delta \varphi)} \]
We can reinterpret this signal using a different pulse shape that accounts for multipath effects.
\[ = G_1 G_2 G_3 \sum_{k=-\infty}^{\infty} a[k] \left[\sum_{l=1}^{L} \alpha_l e^{-j 2 \pi f_1 \tau_l} p(nT_2 - \tau_l - kT_s)\right] e^{j(2 \pi (\Delta f)nT_2 + \Delta \varphi)} \]
In this view, we have taken the pulse shape and convolved it with the effects of the channel. This is the pulse shape the receiver actually sees. Ideally, when using a matched filter, we would match this filter, not the base pulse shape, so as to reduce the effects of non-idealities in our detection.
Nothing about the equation has changed here, but our perception of what is happening is different. It can be useful to think of the process in this way when designing our discrete time signal processing.
