Scope of investigation:
- Modulation Schemes: BPSK, QPSK, and 16QAM
- Pilot-based Channel Estimation Techniques:
- Least Squares(LS)
- Minimum Mean Squared Error(MMSE)
- Linear Minimum Mean Square Error(LMMSE)
- Channel Coding
- (7,4) Hamming Code
- (1,2) Convolutional Code
- Hard decisions using Hamming distance as metrics
The author of this project uses C++(14) and MATLAB(R_2022b). Addtional package manager such as Homebrew is used but not required for program execution. CMake is recommended for simpler building process.
- fftw3: Fast Fourier Transform library
- MATLAB Data API
- Eigen C++: A library for linear algebra
- Google Test: Unit Test
$ mkdir build
$ cd build
$ cmake ../
$ make
$ ./ofdm
The author of this project uses the macOS built-in lldb debugger and AddressSanitizer.
To start debugging, use 'lldb ofdm' after cmake. Use "b ${file_name} : ${line_number}" to set a breakpoint. After setting breakpoints, use "r" to run the debugger.
To set a watchpoint in a loop. Start a process first, and pause at a breakpoint where the watched variable is within scope. Use watch set var ${var_name} to set a watch point. And then add a new condition to this variable using watch modify -c '(${var_name} == ${watch_value})'. Remove the breakpoint within the scope and continue the process. One can call watch list to check existing watchpoints.
- s: step in
- c: continue
- n: step over
- p ${var_name}: print variable
- q: quit debugger
Currently, Google Test is being used to test class functionalities inside the ErrorCorrection Folder. To run existing tests, go to test folder and run ./test.sh to start the tests. You can also add more unit tests to the tests.cpp file.
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MATLAB does column-major reshape operations, while C++ performs row-major operation. Be cautious!
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filter() in ofdm.cpp does convolution by brute force. Truncate the last $(CP_length_) elements to get the desired behavior.
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generateAWGN() now uses std::mt19937() random number generator. This is the key to ensure similar behavior with respect to MATLAB implementation. Using default rng would, for example, results in double symbol errors when running bpsk for 5000 iterations.
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Why would not multiplying channel impulse response (CIR) by sqrt(rho) significantly impact 16qam but not bpsk or qpsk?
Interesting observation: compensating for both mag and phase of a rayleigh channel is the same as compensating for phase only.
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X-axis: Is it SNR or Eb/N0 when average bit energy is the same across different modulation schemes? what is the relationship between SNR and EbN0?
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What is the purpose of cyclic prefix?
Purpose of cyclic prefix: we have managed to restrict the inter-symbol interference to only symbols in the same block and eliminated the effect of inter-block interference. Adding cyclic prefix is essentially a circular convolution between input x and channel impulse response (CIR) h.
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Why is the Multi-carrier modulation signal represented as
$x(t) = \Sigma_k{X_ke^{j2\pi k F_0 t}}$ ?Let us represent the
$k_{th}$ transmitted symbol as$X_k$ and its corresponding subcarrier as$e^{j2\pi kF_0t}$ Hence, the received symbol without noise is$y(t) = \Sigma_{k} X_ke^{j2\pi kF_0t}$ . On the receiver side, to extract the$l_{th}$ symbol,$\tilde{X_l} = F_0 \int_{0}^{1/F_0}e^{-j2\pi kF_0t}y(t)dt$ where$F_0 = \frac{B}{N}$ .It is worth noting that IFFT operation only generates the samples of the actual signal in time domain at specific time stamps, which is sampled at Nyquist rate
$f_s = 2 f_{max}$ . As opposed to MCM, where we need to generate all the symbols and modulators, OFDM replaces all that with simple IFFT and FFT operations. -
Is having multiple pilot subcarriers that span the bandwidth a mean to estimate the channel?
Yes, having multiple pilot subcarriers that span the bandwidth in an OFDM (Orthogonal Frequency Division Multiplexing) system is indeed a common method to estimate the channel's frequency response. These pilot subcarriers serve as known reference points in the transmitted signal, allowing the receiver to estimate how the channel affects the signal at various frequencies within the bandwidth.
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Pilot symbols vs. Pilot subcarriers
In the system that I am using right now, these two terms have different meanings. In a single iteration, 10 ofdm symbols with each symbol spread across 16 separate subcarriers are transmitted through a channel. Pilot symbol corresponds to block-type pilot implementation, while pilot subcarriers (inserting pilot tones) are used in comb-type pilot implementation.
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#9 leads to this question. Does a block-fading model reflect the reality?
In both cases, you need to interpolate the channel between the pilots you've got. Both cases are typically suboptimal, since they'd only work perfectly for (a) actual block-fading (which is a convenient model, but doesn't look like reality) or (b) for a channel that is perfectly interpolatable from just a few points of observation in frequency (but that would imply you have designed an OFDM system with too many subcarrier, and that has other downsides).
- Inadequate Understanding of LS and LMS results in stagnate progress.
Least Squares (LS): LS is a deterministic approach that aims to find the estimate that minimizes the sum of squared errors between the estimated values and the observed data points. It does not assume randomness in the underlying estimated value. Minimum Mean Square Error (MMSE): MMSE, on the other hand, assumes that the underlying estimated value is a random variable. It seeks to minimize the expected value of the mean square error (MSE) between the estimated value and the true value, considering the statistical properties of the random variables involved.
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(Sep.12) Feels like I am already inside a rabbit hole (channel estimation). Still quite confused about ML estimation vs. LS estimation vs. MMSE estiamtion
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Carrier Frequency Offset and compensation?
CFO estimation is the process of determining the frequency offset between the transmitter's carrier frequency and the receiver's local oscillator.
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Error correction Code implementation
Since the ofdm implemented in this project can only transmit 10 ofdm symbols over 16 distinct carriers, coding before the actual symbol transmission will result in no transmission on the last few symbols. For example, if the code rate is 4/7, then only the first 160 * (4/7) data symbols will be transmitted in a single OFDM block.
There are two ways to solve this problem: 1. truncate unsent symbols and only calculate the BER/SER on symbols that were actually transmitted. 2. send the left-over symbols using the next OFDM block. The second method is better as it reflects the reality. However, it would add tremendous complexity to the decoder as the last symbol in a single ofdm block cannot be decoded successfully.
(10 x 16)
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * <---- an OFDM symbol
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
.
.
.
* * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * *
^
|
|
|
one subcarrier
class link: https://nptel.ac.in/courses/117104118 NXP documentation: https://www.nxp.com/docs/en/application-note/AN3059.pdf
Estimation techniques in wireless communication channels
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Least Squares When using a single block-type pilot symbol, LS method is the same as dividing the received pilot symbol
$Y(k)$ by the transmitted pilot symbol$X(k)$ $\hat{H}_{es,LS} = X^{-1}Y$ -
Minimum Mean Squared Error (Least Mean Squre)
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Linear Minimum Mean Squared Error