Theoretical derivation of Maximum Likelihood Estimator for Poisson PDF:

by

Suppose X=(x1,x2,…, xN) are the samples taken from a random distribution whose PDF is parameterized by the parameter $latex \theta $. If the PDF of the underlying parameter satisfies some regularity condition (if the log of the PDF is differentiable) then the likelihood function is given by Here $latex f_N(x_N;\theta) $ is the PDF of … Read more

Maximum Likelihood Estimation (MLE) : Understand with example

by

Key focus: Understand maximum likelihood estimation (MLE) using hands-on example. Know the importance of log likelihood function and its use in estimation problems. Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The core idea behind MLE is to find the parameter values that maximize the likelihood … Read more

Estimator Bias

by

Estimator bias: Systematic deviation from the true value, either consistently overestimating or underestimating the parameter of interest. Estimator Bias: Biased or Unbiased Consider a simple communication system model where a transmitter transmits continuous stream of data samples representing a constant value – ‘A’. The data samples sent via a communication channel gets added with White … Read more

QAM modulation simulation — rectangular constellations in Matlab and Python

by

Digital Modulations 101 — 5 lessons (same on every page in this path) 1. BPSK → 2. QPSK → 3. M-PSK sim → 4. QAM sim (you are here) → 5. EVM Tools: BER vs Eb/N0 · EVM → SNR · Eb/N0 ↔ SNR Also on the Matlab → Python path (lesson 3 of 4): … Read more

M-PSK simulation — Matlab and Python BER curves for higher-order phase shift keying

by

Digital Modulations 101 — 5 lessons (same on every page in this path) 1. BPSK → 2. QPSK → 3. M-PSK sim (you are here) → 4. QAM sim → 5. EVM Tools: BER vs Eb/N0 · EVM → SNR · Eb/N0 ↔ SNR Also on the Matlab → Python path (lesson 4 of 4): … Read more

Derive BPSK BER – optimum receiver in AWGN channel

by

BPSK Lab path:Derive BER (theory) → AWGN noise model → BER simulation (Python & Matlab)Tools: BER vs Eb/N0 calculator · Shannon capacity Key focus: Derive BPSK BER (bit error rate) for optimum receiver in AWGN channel. Explained intuitively step by step. BPSK modulation is the simplest of all the M-PSK techniques. An insight into the … Read more

Estimation Theory : an introduction

by

Key focus: Understand the basics of estimation theory with a simple example in communication systems. Know how to assess the performance of an estimator. A simple estimation problem : DSB-AM receiver In Double Side Band – Amplitude Modulation (DSB-AM), the desired message is amplitude modulated over a carrier of frequency f0. The following discussion is … Read more

Derivation of expression for a Gaussian Filter with 3 dB bandwidth

by

In GMSK modulation (used in GSM and DECT standard), a GMSK signal is generated by shaping the information bits in NRZ format through a Gaussian Filter. The filtered pulses are then frequency modulated to yield the GMSK signal. GMSK modulation is quite insensitive to non-linearities of power amplifier and is robust to fading effects. But … Read more

Sampling Theorem – Bandpass or Intermediate or Under Sampling

by

Prerequisite: Sampling theorem – baseband sampling Intermediate Sampling or Under-Sampling A signal is a bandpass signal if we can fit all its frequency content inside a bandwidth $latex F_b $. Bandwidth is simply the difference between the lowest and the highest frequency present in the signal. “In order for a faithful reproduction and reconstruction of … Read more