Region where likelihood function is non-zero

Maximum Likelihood Estimation of Clock Offset

When I started my PhD, one of the first papers I read was On Maximum Likelihood Estimation of Clock Offset by Daniel Jeske [1] from University of California, Riverside. It eventually set the direction of my future research and ultimately my PhD dissertation. I found this paper quite interesting as it talked about the estimation of clock phase offset. Later I went on to explore what was missing here (the clock frequency offset) and more. Keep in mind that carrier phase estimation is a different problem that has already been discussed in the past here, here and here. Most of

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Two grating lobes on the sides

How Do Beams Look Like?

In the article on beamforming, we discussed the interaction of the electromagnetic waves with the antenna array without any description of what the beam shape looks like. As we explore below now, the beam shape is given by the Fourier Transform of individual antenna intensities but the reason behind this is not always explained in most of the textbooks and tutorials on this topic. Where exactly does the Fourier Transform, a conversion tool from time $t$ to frequency $\omega=2\pi F$ domain, come into the picture? And how does the frequency $\omega$ for time domain correspond to phase shift $u$ of

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Linear interpolation between pilot subcarriers

Channel Estimation in OFDM Systems

Channel estimation in single-carrier systems has been described in a previous article. In OFDM systems, each subcarrier acts as an independent channel as long as there is no Inter-Carrier Interference (ICI) left in the synchronized signal. The options of both a training sequence and individual pilots are available for channel estimation and the choice between the two depends on time variation rate of the channel as well as the computational complexity. Many systems acquire the channel through the preamble while employ the pilots for channel tracking. The discussion in this article is mostly based on Ref. [1]. For a simplified

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Error Vector Magnitude (EVM) for 1 symbol

What is Error Vector Magnitude (EVM)?

Measuring the performance of a digital communication system is not a straightforward task as different impairments have different impacts on the final bit error rate. Error Vector Magnitude (EVM) is a useful metric that helps gauge the impact of all impairments simultaneously from a single value. A Single Modulation Symbol We start with observing a single modulation point at the receive end. Once we establish the baseline error in this scenario, we will combine the effect of all such symbol points into a single number. Assume that a modulation symbol S is represented by a blue constellation point in the

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Working of an Early-Late TED

On the Link Between Gardner Timing Error Detector and Early-Late Timing Error Detector

This post is written on an advanced topic mainly for practitioners and researchers in the design of wireless systems. For learning about wireless communication systems from a DSP perspective (the idea behind SDRs), I recommend you have a look at my book. F. M. Gardner described his well known Timing Error Detector (TED) — known as Gardner TED — in his often cited article [1]. Gardner was a pioneer in the area of synchronization and Phase Locked Loops (PLL). Later, M. Oerder (a student of Heinrich Meyr) derived this scheme from the maximum likelihood principle in [2]. Heinrich Meyr is

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