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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Time and frequency response of a lowpass FIR filter designed with Parks-McClellan algorithm for N=33

Why FIR Filters have Linear Phase

One of the most attractive properties of a Finite Impulse Response (FIR) filter is that a linear phase response is easier to achieve. Not all FIR filters have linear phase though. This is only possible when the coefficients or taps of the filter are symmetric or anti-symmetric around a point. Today I want to describe the reason behind this kind of phase response in an intuitive manner. We have described Finite Impulse Response (FIR) filters before. Moreover, we have also discussed that the Discrete Fourier Transform (DFT) of a signal is complex in general and therefore both magnitude and phase

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A complex number

Convolution and Fourier Transform

One question I am frequently asked is regarding the definition of Fourier Transform. A continuous-time Fourier Transform for time domain signal $x(t)$ is defined as \[ X(\Omega) = \int _{-\infty} ^{\infty} x(t) e^{-j\Omega t} dt \] where $\Omega$ is the continuous frequency. A corresponding definition for discrete-time Fourier Transform for a discrete-time signal $x[n]$ is given by \[ X(e^{j\omega}) = \sum _{n=-\infty} ^{\infty} x[n] e^{-j\omega n} \] where $\omega$ is the discrete frequency related to continuous frequency $\Omega$ through the relation $\omega = \Omega T_S$ if the sample time is denoted by $T_S$. Focusing on the discrete-time Fourier Transform for

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