Digital Signal Processing

Project overview

I developed digital filters based on square-root raised cosine FIR filter for QPSK, QAM and OFDM modulations in MATLAB. The modulations mentioned are high-frequency signals used in wireless communication. The filters allow to narrow the spectrum of the signal without the data loss. That can increase the speed for OFDM modulation and decrease the cost for QPKS or QAM modulations.

Business Value

When you build an embedded system or device that must communicate wirelessly on big distance, you have to register the frequency band you are going to use and pay accordingly to the band-width you use.

Therefore, if you can transmit the same data for the same time, but occupying less band width you can save money with more efficient transmission. That is how digital filters for wireless transmission can help.

Another value of applying digital filters to modulated signals is that you can increase the data transfer with OFDM modulation. This modulation works in a way that you send a pack of QAM signals together. However, if you narrow their spectrum then you can put more QAM signals together and transmit more bits per second which increases the speed of communication.

Technical details

Here you can find explanation of the work principle. However, to explain the work of filtering, the modulation signals must be explained as well.

Let’s take QAM-16 as an example. It consist of the sum of Imaginary part and Real part both multiplied by high-frequency oscillations.

s = Re * cos(wt) - Im * sin(wt)

Therefore, the signal can be shown on complex plane neglecting the high frequency oscillation. In the Figure below you can see 4 stars on the plane corresponding to 4 unique values of QAM-4 and carrying 2 bits each.

The signal that carry arbitrary bit flow using could be shown as plots of the Imaginary and Real parts like in Figure below

In reality, bits are not transmitted one right after another, as the transmission and processing is the time process. Therefore, there is a sampling frequency when the signal is measured and bits are interpreted. In Figure below you can see the same Re and Im values as above, but with time pauses between two symbols corresponding to sampling period.

These pikes correspond to the same values from the Figure above it, but they just have pauses between values. It comes out that when you fill pauses with the same value as previous symbol, you get simple digital square signal which is easy to build, but it gives a huge spectrum shaped as sin(x)/x. Figures below show square-like signal and its corresponding very wide spectrum.

However, there is a way to narrow this spectrum with square-root raised cosine FIR filter. You can simply feed the spike-like signal to it and it will fill the values between spikes in a way to decrease the used spectrum. Your final signal and spectrum will look like:

Reasonable question may arise asking if the transmitted symbols are the same? This is important to check to ensure there is no data loss with such data manipulation.  To check it I can fold the signal by length of 4 symbols, so I see overlayed signals representing 4 symbols again and again. In that way I can locate the positions of sampling, there the values must be one of given from QAM-4 Complex Plane, while all other values are arbitrary. That is called Eye-Diagram.

It is clearly seen that this eye-diagram has 4 nodes on the bottom and on the top and their Y-value corresponds to values from Complex Plane. The lines between each node represents how signal connects one symbol with another after passing cosine filter. This exact “arbitriarity” allows to narrow the spectrum with no data loss.

Now many frequency-shifter QAM-4 signals can be put together forming an OFDM signal that is currently used as main modulation in Wi-Fi data transfer.

Results

I created a pipeline with square-root raised cosine FIR filter to reduce occupied spectrum by QPSK or QAM signal. That can both save money on occupied frequency band-width and can allow OFDB signal to put more QAM modulated signals together to deliver more information per second, improving the speed of wireless communication.