Plots for positive integer powers of x in 3D
DSP / Miscellaneous

A Real-Imaginative Guide to Complex Numbers

June 18, 2020 On a cold morning in August 2015, I narrowly missed a train to my office in Melbourne city. With nothing else to do in the next 20 minutes, my mind wandered towards an intuitive view of complex numbers, something that has puzzled me since long. In particular, I wanted to seek answers to the following questions. (a) What is the role of the number $\sqrt{-1}$ in mathematics? What sets it apart from other impossible numbers, e.g., a number $k$ such that $|k|=-1$? (The origins of this question might lie in how I cut apple slices for my

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Wireless/SDR

Early-Late Bit Synchronizer in Digital Communication

In an article on Phase Locked Loop (PLL) for symbol timing recovery, we described an intuitive view of a maximum likelihood Timing Error Detector (TED). We saw that the timing matched filter is constructed by computing the derivative of the matched filter and consequently its output is the derivative of the input signal. Naturally, this output is more fine-grained and hence accurate when the number of samples/symbol $L$ is large. Here, $L$ must be several times larger than the minimum limit set by the Nyquist theorem. However, in most applications, reducing the complexity of the timing locked loop is far

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Two way transmission
Localization

The Big Picture of Localization

Digital Signal Processing (DSP) enables us to find the range of a device transmitting a wireless signal with a particular structure under some conditions. To understand how this process works, we need to look at the big picture of a localization process. Localization implies locating the unknown position of a source which can be computed in a straightforward manner if its ranges from some reference nodes can be found. Various techniques are employed for this purpose, some of which are Received Signal Strength Indicator (RSSI), time of arrival, time difference of arrival and angle of arrival. Phase of arrival is

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A V-BLAST architecture for 4 Tx antennas
Wireless/SDR

V-BLAST with Successive Interference Cancelation

In the article on Zero-Forcing detector for MIMO receivers, we have seen that the performance of linear detectors is unsatisfactory for actual implementations of conventional MIMO systems. Their main attraction comes from their low computational complexity. To strike a nice balance between performance and complexity, a neat trick is employed by the algorithm known as Successive Interference Cancelation (SIC). The concept was devised by Gerard Foschini from Bell Labs, although it was not a new idea. Successive interference cancelation was already proposed for the detection algorithms in CDMA systems. Again, the fundamental idea was borrowed from decision feedback equalization schemes

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Logic behind Mueller Muller TED
Wireless/SDR

Mueller and Muller Timing Synchronization Algorithm

Proposed in 1976, Mueller and Muller algorithm is a timing synchronization technique that operates at symbol rate, as opposed to most other synchronization algorithms that require at least 2 samples/symbol such as early-late and Gardner timing error detectors. All of these are feedback techniques that operate within a PLL. Feedforward methods such as digital filter and square timing synchronization are also feasible due to powerful digital signal processing that avoids feedback problems such as hangups. The most confusing thing communication engineers and radio hobbyists find about Mueller and Muller algorithm algorithm is the cross product in its expression: matched filter

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