Starting with simple channel prediction

Previously I wrote about the application of CP in OFDM, mainly to record the idea of 1.0 stars. Today I suddenly want to write something about channel characteristics. The reasons are as follows:

1) the status of the channel in simulation is unquestionable. Simulation under different channels is the focus of many topics, and it is almost unknown.

2) Since there is no entry point, we can only start with the simplest channel prediction. Of course we will also talk about the CP issue, after all, the impact of simulation CP on OFDM inspires you to look at the relevant materials.

3) another reason is that writing an article is a process of thinking.

4) At last, Mark's wisdom tooth leaves me, hurting ing!

Let's look back at some tips:

1) when the signal passes through multiple paths, the amplitude and phase of each path are random. We know that the amplitude is subject to the Fourier distribution, and the phase is even, the typical Jakes model and various improved algorithms are used for simulation (the frequency shift is taken into account ).

2) channel models in wireless environments are often multi-path (large diameter) (direct diameter, reflection, scattering, etc.), because the multi-path leads to frequency selectivity, which is called frequency selectivity, that is, the channel has different effects on signal components of different frequencies.

3) In addition, assuming that the moving conditions are added, our channel is time-varying, and time-varying will lead to frequency domain dispersion, also known as the frequency shift.

4) Therefore, we always model a wireless channel as a linear time-varying channel.

Starting from the simplest multi-path channel, we simulate two paths, each of which has no fading, which is a fixed weighted value. Let's take a look at the following example:

Next, we will use a program to verify:

clear allclose alltx_data = [1 0 0 0];data_delay = [0 1 0 0];rx_data = tx_data + data_delay;fft_tx = fft(tx_data)fft_rx = fft(rx_data)


The result is as follows:
In fact, the above program verification is very easy. If you change the data to [1 2 3 4], it is definitely not the result. What is the problem? This starts with formula (2). What sort of DFT can produce such a result? Looking back at the DSP, we can know that the cyclic shift, that is, the periodic shift, can produce the effect of formula (2). Why does the above program have no cyclic shift and the correct result? This is because the above data [1 0 0 0 0] linear shift and periodic shift have the same effect. To verify that we can change the above data, we 'd better take the innocent [1, 2, 3, 4] for a look.
clear allclose alltx_data = [1 2 3 4];data_delay_linear = [0 1 2 3];data_delay_period = [4 1 2 3];rx_data_linear = tx_data + data_delay_linear;rx_data_period = tx_data + data_delay_period;fft_tx = fft(tx_data)fft_rx_linear = fft(rx_data_linear)fft_rx_period = fft(rx_data_period)


The result is as follows:
Obviously, only the result after cyclic shift superposition can satisfy the general channel characteristics, such as K = 1, (-2 + 2I) * H (1) = (-2 + 2I) * (1-I) = 4I. We only need to obtain the values of H (k) at the receiving end, so it is very easy to compensate for the loss caused by the channel.
Well, let's go back to OFDM. We place the Constellation-mapped symbols on different ifft_bin, that is, they are sent at different frequencies, in the previous blog article, we have explained in the application of circular prefixes in OFDM (I) that the channel is only weighted by H (k) for each frequency, in this case, we do not need to use such complex balancing technology. We only need to predict the value of H (K), and then the inverse operation can restore the original symbol, we used the cyclic prefix technology to achieve the above results. In that blog, we proved from the cyclical prefix to linear convolution, but it does not explain why the change was successful. I hope to understand it intuitively today. Now, with the simple example above, we can see a little clue. It is also explained by the innocent [1 2 3 4] (here, the CP length is 3 ):
From the above figure, we can see that we can change the linear shift to the cyclic shift within the interval [] only when the channel Impulse Response length is smaller than the CP length, therefore, after the CP signal is added, after the channel transmission, we will remove the CP length from the receiver. This is not intuitive enough. Why does the sender add CP, what if the receiver removes CP to implement cyclic convolution? Well, to solve the problem, let's take a look at what is cyclic convolution and linear convolution. For an intuitive understanding, I don't plan to use the theory on DSP, in a more intuitive way.
In fact, we can conclude that, from a macro perspective, our signal must undergo linear convolution with the channel's impulse response, just like the above example of innocent [1 2 3 4], the weighted sum of each latency version number (here all the weights are 1), but when we look at it from a local perspective, we don't just care about the interval [], so a very obvious fact is in front of us: Here we are doing circular convolution.
Now, our thoughts are almost the same. To sum up, we map the constellation symbol x = [X1, x2 ..., XN] After IFFT operation, the time domain signal X = [X1, x2 ..., XN], we add CP and turn it into x', turning it into a 'macro 'signal going through the channel. In general, linear convolution is correct! However, the original time-domain signal X always maintains cyclic convolution (only the CP length must be greater than the channel Impulse Response length ), then we can use the means to obtain the channel feature H (k). Each K is for the symbol XK on each sub-carrier, therefore, a frequency-selective channel becomes multiple flat channels.
Note: A simple sentence in the book: CP converts linear convolution into cyclic convolution. In fact, I think a better understanding is:Macro linear convolution becomes local cyclic convolution,After all, the receiver will remove the CP, that is to say, we are always concerned about the local part. Because the range is obtained by our IFFT transformation, the CP is to reduce the complexity of the receiver at the cost of the channel bandwidth.