In a previous post we have seen that MIMO fading capacity is much higher than AWGN capacity with multiple antennas. How is this possible? How can randomness added by a fading channel help us? In this post we try to find the reason for this. Let’s assume the following signal model for a Multi Input Multi Output antenna system.
Here s is the NT by 1 signal vector, w is the NR by 1 noise vector and H is the NR by NT channel matrix. The received signal vector is represented by x which has dimensions of NR by 1. In expanded form this can be written as (assuming NT =4 and NR =4):
In a previous post we had discussed MIMO capacity in a fading environment and compared it to AWGN capacity. It sometimes feels unintuitive that fading capacity can be higher than AWGN capacity. If a signal is continuously fluctuating how is it possible that we are able to have reliable communication. But this is the remarkable feature of MIMO systems that they are able to achieve blazing speeds over an unreliable channel, at least theoretically. It has been shown mathematically that an NxN MIMO channel is equivalent to N SISO channels in parallel.
Before we delve deep into Minimum Shift Keying (MSK) and its performance in presence of co-channel interference the reader is advised to look at the following posts.
Post 1 – MSK BER performance in AWGN and flat fading environment when viewed as extension of BPSK
Post 2 – MSK Power Spectral Density and its BER performance in AWGN when viewed as a CPM
Post 3 – MSK BER Performance in AWGN and flat fading environment when viewed as a CPM
Co-channel interference is a phenomenon widely encountered in wireless communication systems and the main reason for that is frequency reuse, which allows the same frequency band to be used over and over again in geographically non-contiguous areas. GSM and other wireless communication systems, using MSK modulation, suffer from the same problem. This has been widely studied in the literature and interference rejection techniques have been proposed. The worst case is one where the power of both the signals (wanted signal and interference) is almost the same and there is no frequency or phase offset.
I – In the previous post we presented the mathematical model and code for BER calculation of a popular modulation scheme called MSK. However in the code we shared, we only considered one sample per symbol, which makes MSK look like BPSK. While BPSK symbols fall on the real axis, MSK symbols alternate between real and imaginary axes, progressing by π/2 phase during each symbol period. MSK signal thus has memory and this can help in demodulation using advanced techniques such as Viterbi Algorithm.
I - Minimum Shift Keying (MSK) is a type of Continuous Phase Modulation (CPM) that has been used in many wireless communication systems. To be more precise it is Continuous Phase Frequency Shift Keying (CPFSK) with two frequencies f1 and f2. The frequency separation between the two tones is the minimum allowable while maintaining orthogonality and is equal to half the bit rate (or symbol rate, as both are the same). The frequency deviation is then given as Δf=Rb/4. The two tones have frequencies of fc±Δf where fc is the carrier frequency. MSK is sometimes also visualized as Offset QPSK (OQPSK) but we will not go into its details here.
When a wireless signal travels from a transmitter (Tx) to a receiver (Rx) it undergoes some changes. In simple terms the signal s(t) is scaled by a factor h(t) and noise n(t) is added at the receiver. Let’s take this discussion forward with a simple example. Suppose the Tx transmits one of two possible symbols, +1 or -1. In technical lingo this is called Binary Phase Shift Keying (BPSK). If the channel scaling factor is 0.1 we will either get a +0.1 or -0.1 at the Rx to which AWGN noise is added. The noise is random in nature (having a Gaussian distribution) but for simplicity we assume that it can have one of two values, +0.01 or -0.01.