ECE414 Wireless Communications

1 Abstract

This is a review of a course, ECE414 wireless communications. We learn the terminology used by the people building communication systems, some which is quite old (~1800s), to describe how they pass messages over great distances.

Then we learn about the statistical models used to characterize the likelihood of different things happening to our message, which is passed as EM waves over great distances.

2 Description of the Model

ECE414 at its core is a model of a real situation. We have two people who wish to communicate with each other. In all of the situations one will be talking, while the other is listening. The speaker is called the transmitter (TX) whereas the listener is called the receiver (RX).

Sometimes one of the people will be moving. In this case they are also called a mobile station. Sometimes they will be stationary. Then they are called a base station.

The communication between TX and RX must happen in a space. Sometimes the space is empty and TX and RX can see each other; in this case the communication is line of sight. Sometimes the space is blocked by walls and RX can only hear echoes. No matter what, the space in which the communication happens is named the channel.

This is the problem we are facing.

3 The difficulties

1. To talk over great distances, we cannot just shout. We convert what we want to say using a code of symbols which in complete form is an alphabet. Then we use circuits to output signals such as radio waves which can travel the great distance (the channel). This is called modulation. More circuits are used on the other side to get the original symbols back. This is called demodulation.
2. The environment is a hostile propagation medium: it is very hard to shout and be heard on the other side. We are in a noisy enviornment, and what we say will be heard with many other voices at the same time. To deal with this, we:
   1. Use error-correcting codes such that even if a particular symbol is interpreted incorrectly, we still have a chance of getting the right message.
   2. Design the system such that the TX signal power is greater than the expected noise by some threshold.

4 Our channel

Let us create a mathematical model of the situation described above. This will give us intuition of what is happening. We must remember though, that real circuits must be designed for us to realize any design. So the math is a design language but isn’t the whole reality.

The model is called the multi-path fading channel model and the key signal processing concept is called the linear time-variant filter. The filter is our channel model.

Why use such mathematics?

1. By modelling the system using a filter, the whole channel becomes a system. There is a lot of analytical methods on analyzing such systems and such filters: for example, we can model the whole communication by putting the filter in cascade with other filters. We will see more later.

The filter looks like this:

$$h(\tau, t) = \sum_{i=1}^N(t) \alpha_i(t) c(\phi_{D_i}(t))\delta(\tau-\tau_i(t))$$

I present the filter before the physical model as I follow along better. What is this describing in reality?

The received signal r(t) will be equal to the sum of the amplitudes of the line of sight path (LOS) signal and each resolvable multipath component: each of these signals may have a shift in phase (Doppler phase shift) Φ_Dn, change in amplitude (fading/attenuation) α, and time delay τ_n due to the path travelled.

$$h(\tau, t) = \sum_i^N \alpha_i(t) c(2\pi f_c(t - \tau_i(t)) + \Phi_{D_i}(t))$$

Finally, the received signal r(t) at RX from the model is

r(t) = h(τ, t) ⊛ s(t)

I am writing the original baseband (also named low-pass) signal as s and the carrier signal c, both a function of time. The process of modulation itself is when a carrier signal is multipled by the signal we want to transmit and it is the math which represents the conversion of the original speech into radio/ EM waves which can travel far greater distances (this multiplication is done on the circuits).

Note the interresting things about this model:

1. The end result is a sum: this means that the contributions by each component are linear. This is because of how EM signals themselves work physically: they can be modelled (exactly to our observations) using periodic e^j2πf_ct and are linear.
2. The attenuation α_i for each multipath component is described as a fraction of the original signal strength: that is, the shape of the transmitted signal does not change, it is still the same function.
3. The delay is applied equally to both the original signal and the modulating carrier function, but the doppler phase shift is only applied to the carrier.

The LTV (linear time varying) filter describes a unique situation where the impulse is applied to its input at an initial time, but the other time variable describes the instant of observing the output (final time). Both must be the same for the same output to be received. In this case, the received time (where the signal is at RX) is t, while the time when the signal is launched into the channel is t − τ: τ is the delay.

4.1 “Resolvable multi-path components”

So then the sum is of resolvable multi-path components: what are these components? These represent reflections of the same communicated signal over a great distance. When we are broadcasting a signal from TX to RX, this is a broadcast that happens in a circle (it goes in all directions) but only RX interprets it: just as if we speak to one another, other people may also hear our speech, but we hope will just ignore it. If you speak in a large room, you may hear echoes of your own voice. These are the auditory equivalents of the reflections that happen to produce multi-path components.

For the multipath-component, we are assuming that many of these reflections are received by RX at the same time, forming a continuous signal. This is why our function reconstructs the received signal as a sum of all the reflections. Each of these components being a reflection travels a different distance. This is represented by each component having its own delay τ_i. It may also go through different environments. Therefore, each component has its own attenuation α_i and doppler shift Φ_Di.

If we were considering every single ray or reflection our calculations would be never ending. A resolvable multi-path component is then one representative signal for many, many reflections with similar delay. Two signals are non-resolvable (can be merged into a single resolvable multipath component) if they meet a certain empirical criteria:

|τ₁ − τ₂| >  > τ_u

Where τ₁, τ₂ are the signal delays for each signal, and τ_s is the signal delay.

4.2 Analysis in the frequency domain

We may now define the frequency transfer function for the channel. This is

H(f, t) = 𝔽_τ{h(τ, t)}

Essentially, we are fixing the delay of the channel and replacing the delay with a frequency parameter (sweeping the possible delays as frequencies) while the received time stays as a variable.

$$H(f, t) = \int\limits^{\infty}_{-\infty} h(\tau, t) c(2\pi f_c \tau) d\tau$$

Then R(f, t) = H(f, t)S(f)