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# DSPLOG

## Sitemaps
[XML Sitemap](https://dsplog.com/sitemap_index.xml): Includes all crawlable and indexable pages.

## Posts
- [Positional Encoding in Transformers](https://dsplog.com/2026/07/04/positional-encoding-in-transformers/): In the seminal paper Attention is All you Need (Vaswani et al 2017), the authors proposed Transformer architecture where all tokens in sequence can be processed in parallel. As the architecture process all tokens simultaneously, the concept of positional embeddings to encode the sequence information is needed. In this post, we cover few positional encoding techniques and techniques for extending the pre-trained context length.
- [Loss functions for handling class imbalance](https://dsplog.com/2026/03/05/loss-functions-for-handling-class-imbalance/): For handling class imbalance, multiple stratetgies have emerged. Post covers Weighted Cross Entropy, Focal Loss, Assymetric Loss, Class Balanced Loss and Logit Adjusted Loss.
- [Word Embeddings using neural networks](https://dsplog.com/2025/12/27/word-embeddings-using-neural-networks/): The post covers various neural network based word embedding models. Starting from the Neural Probabilistic Language Model from Bengio et al 2003, then reduction of complexity using Hierarchical softmax and Noise Contrastive Estimation. Further works like CBoW, GlOVe, Skip Gram and Negative Sampling which helped to train on much higher data.
- [Gradients for multi class classification with Softmax](https://dsplog.com/2025/06/22/gradients-for-multi-class-classification-with-softmax/): In a multi class classification problem, the output (also called the label or class) takes a finite set of discrete values . In this post, system model for a multi class classification with a linear layer followed by softmax layer is defined. The softmax function transforms the output of a linear layer into values lying between 0 and 1, which can be interpreted as probability scores.
- [Gradients for Binary Classification with Sigmoid](https://dsplog.com/2025/05/17/gradients-for-binary-classification/): In a classification problem, the output (also called the label or class) takes a small number of discrete values rather than continuous values. For a simple binary classification problem, where output takes only two discrete values : 0 or 1, the sigmoid function can be used to transform the output of a linear regression model into a value between 0 and 1, squashing the continuous prediction into a probability-like score. This score can then be interpreted as the likelihood of the output being class 1, with a threshold (commonly 0.5) used to decide between class 0 and class 1.
- [Gradients for linear regression](https://dsplog.com/2025/05/01/gradients-for-linear-regression/): Understanding gradients is essential in machine learning, as they indicate the direction and rate of change in the loss function with respect to model parameters. This post covers the gradients for the vanilla Linear Regression case taking two loss functions Mean Square Error (MSE) and Mean Absolute Error (MAE) as examples.
- [Migrated to Amazon EC2 instance (from shared hosting)](https://dsplog.com/2013/03/11/migrated-to-amazon-ec2-from-shared-hosting/): Being not too happy with the speed of the shared hosting, decided to move the blog to an Amazon Elastic Compute Cloud (Amazon EC2) instance.  Given this is a baby step, picked up a micro instance running an Ubuntu server and installed Apache web server, MySQL, PHP . After doing a bit of tweaking with this new instance, imported the SQL database and other files from the shared hosting and pointed the A name record to the new IP address. This switch happened over this weekend.
- [GATE-2012 ECE Q28 (electromagnetics)](https://dsplog.com/2013/02/20/gate-2012-ece-q28-electromagnetics/): Question 28 on electromagnetics from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [Image Rejection Ratio (IMRR) with transmit IQ gain/phase imbalance](https://dsplog.com/2013/01/31/imrr-transmit-iq-gain-phase-imbalance/): The post on IQ imbalance in transmitter, briefly discussed the effect of amplitude and phase imbalance and also showed that IQ imbalance results in spectrum at the image frequency. In this article, we will quantify the power of the image with respect to the desired tone (also known as IMage Rejection Ratio IMRR) for different values of gain and phase imbalance.
- [GATE-2012 ECE Q15 (communication)](https://dsplog.com/2013/01/25/gate-2012-ece-q15-communication/): Question 15 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q7 (digital)](https://dsplog.com/2013/01/23/gate-2012-ece-q7-digital/): Question 7 on digital from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q13 (circuits)](https://dsplog.com/2013/01/22/gate-2012-ece-q13-circuits/): Question 13 on analog electronics from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q34 (signals)](https://dsplog.com/2012/12/29/gate-2012-ece-q34-signals/): Question 34 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q12 (math)](https://dsplog.com/2012/12/28/gate-2012-ece-q12-math/): Question 12 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q11 (signals)](https://dsplog.com/2012/12/27/gate-2012-ece-q11-signals/): Question 11 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q2 (communication)](https://dsplog.com/2012/12/15/gate-2012-ece-q2-communication/): Question 52 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q52 (electromagnetics)](https://dsplog.com/2012/12/01/gate-2012-ece-q52-electromagnetics/): Question 52 on electromagnetics from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q16 (electromagnetics)](https://dsplog.com/2012/11/25/gate-2012-ece-q16-electromagnetics/): Question 16 on electromagnetics from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q39 (communication)](https://dsplog.com/2012/11/17/gate-2012-ece-q39-communication/): Question 39 on communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q47 (math)](https://dsplog.com/2012/11/13/gate-2012-ece-q47-math/): Question 47 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [Update : Correction to solution of GATE-2012 ECE Q38](https://dsplog.com/2012/11/09/update-correction-gate-2012-ece-q38/): Thanks to Mr. Raghava G D's comments on the post discussing Question 38 on Communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper, realized that I had made an error in the solution. Have the updated the post with the right answer and additional explanations.
- [GATE-2012 ECE Q38 (communication)](https://dsplog.com/2012/11/03/gate-2012-ece-q38-communication/): Question 38 on Communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q3 (communication)](https://dsplog.com/2012/11/01/gate-2012-ece-q3-communication/): Question 3 on Communication from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q26 (electronic devices)](https://dsplog.com/2012/10/27/gate-2012-ece-q26-electronic-devices/): Question 26 on Electronic Devices from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q36 (math)](https://dsplog.com/2012/10/24/gate-2012-ece-q36-math/): Question 36 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q46 (math)](https://dsplog.com/2012/10/23/gate-2012-ece-q46-math/): Question 46 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q24 (math)](https://dsplog.com/2012/10/14/gate-2012-ece-q24-math/): Question 24 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q25 (math)](https://dsplog.com/2012/10/09/gate-2012-ece-q25-math/): Question 25 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [GATE-2012 ECE Q6 (digital)](https://dsplog.com/2012/10/07/gate-2012-ece-q6-digital/): Question 6 on digital circuit from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
- [Modeling phase noise (frequency domain approach)](https://dsplog.com/2012/09/30/modeling-phase-noise-frequency-domain/): In typical wireless system simulations, there is  a need to model the phase noise profile of the local oscillator. For eg, the phase noise profile of the oscillator can be of the shape described in the post on Phase Noise Power Spectral Density to Jitter. While looking around for example Matlab code, found two references which uses the approach of defining the phase noise profile in frequency domain, and then using ifft() to convert to the time domain samples. This post gives a brief overview of the modeling and provides an example Matlab/Octave code.
- [Migration to new template (skin)](https://dsplog.com/2012/09/11/migration-to-new-template-skin/): Hi,
- [Transmit spectrum with phase noise](https://dsplog.com/2012/08/28/transmit-spectrum-phase-noise/): The earlier posts on phase noise discussed about phase noise in oscillators, conversion of phase noise profile to jitter and the impact of phase noise on the error vector magnitude (evm). This post discuss the impact of phase noise on the spectrum of the transmit waveform.  A simple random QPSK modulated symbols, oversampled and passed through a root raised cosine filtering is used for the simulation.
- [EVM with phase noise](https://dsplog.com/2012/07/09/evm-phase-noise/): The previous post on phase noise discussed about finding the root mean square phase noise for a given phase noise profile. In this post let us discuss about the impact of phase noise on the error vector magnitude (evm) of a transmit symbol.
- [Phase noise power spectral density to Jitter](https://dsplog.com/2012/06/22/phase-noise-psd-to-jitter/): Following a brief discussion with my friend Mr. Rethnakaran Pulikkoonattu on phase noise profiles, he pointed me to his write up on Oscillator Phase Noise and Sampling Clock Jitter . In this post, we will discuss the math behind integrating the phase noise power spectral density (in dBc/Hz) to find the root mean square jitter value.
- [Oscillator phase noise](https://dsplog.com/2012/06/08/oscillator-phase-noise/): Oscillators are used in typical radio circuits to drive the mixer used for the up-conversion or down-conversion of the passband transmission. Ideally, the spectrum of the oscillator is expected to have an impulse at the frequency of oscillation with no frequency components else where. However the spectrum of practical oscillators do have spectrum skirts around the oscillation frequency caused due to phase noise. This post discuss about the phase noise of oscillator and the metrics used to specify it.
- [Thermal noise of RC low pass filter](https://dsplog.com/2012/05/06/thermal-noise-rc-low-pass-filter/): This post discuss about the thermal noise in RC low pass filter. Using the noise equivalent model using resistor with a voltage source, which gets passed through a no noise RC low pass filter. The noise power at the output is computed by integrating the output voltage spectral density over all frequencies.
- [Noise Figure of cascaded stages](https://dsplog.com/2012/04/21/noise-figure-cascaded-stages/): Following the discussion on thermal noise and it's modeling and noise figure computation for a simple resistor network, in this article let us discuss the Noise Figure of cascaded stages.
- [Noise Figure of resistor network](https://dsplog.com/2012/04/13/noise-figure-resistor/): The post on thermal noise described the noise produced by resistor  ohms over bandwidth  at temperature Kelvin. In this post, let us define the noise voltage at the input and output of a resistor network and further use it to define the Noise Figure of such a network.
- [Solved!](https://dsplog.com/2012/04/04/solved-rubiks-cube/): After 6 months, 2 cube's and countless twists and turns, extremely glad to reach here.
- [Thermal Noise and AWGN](https://dsplog.com/2012/03/25/thermal-noise-awgn/): A friend called me up couple of days back with the question - How white is AWGN? I gave him an answer over phone, which he was not too happy about. That got me thinking bit more on the topic and the result is this post - brief write up on thermal noise and it's modelling as Additive White Gaussian Noise aka AWGN.
- [Hamming (7,4) code with soft and hard decoding](https://dsplog.com/2012/03/15/hamming-code-soft-hard-decode/): An earlier post we discussed hard decision decoding for a Hamming (7,4) code and simulated the the bit error rate. In this post, let us focus on the soft decision decoding for the Hamming (7,4) code, and quantify the bounds in the performance gain.
- [ADC SNR with clock jitter and quantization noise](https://dsplog.com/2012/02/22/adc-snr-clock-jitter-quantization-noise/): My friend and colleague Mr. Vineet Srivastava pointed me to a nice article on  clock jitter - Clock Jitter Effects on Sampling : A tutorial - by Carlos Azeredo-Leme, IEEE Circuits and Systems Magazine, Third Quarter 2011. In this post, let us discuss the total Signal to Noise Ratio at the output of an analog to digital converter (ADC) accounting for errors due to sampling clock jitter and quantization noise.
- [Weighted Least Squares and locally weighted linear regression](https://dsplog.com/2012/02/05/weighted-least-squares-and-locally-weighted-linear-regression/): From the post on Closed Form Solution for Linear regression, we computed the parameter vector  which minimizes the square of the error between the predicted value  and the actual output  for all  values in the training set. In that model all the  values in the training set is given equal importance.  Let us consider the case where it is known some observations are important than the other. This post attempts to the discuss the case where some observations need to be given more weights than others (also known as weighted least squares).
- [Least Squares in Gaussian Noise – Maximum Likelihood](https://dsplog.com/2012/01/15/least-squares-gaussian-noise-maximum-likelihood/): From the previous posts on Linear Regression (using Batch Gradient descent, Stochastic Gradient Descent, Closed form solution), we discussed couple of different ways to estimate the  parameter vector in the least square error sense for the given training set. However, how does the least square error criterion work when the training set is corrupted by noise? In this post, let us discuss the case where training set is corrupted by Gaussian noise.
- [Symbol Error rate for QAM (16, 64, 256,.., M-QAM)](https://dsplog.com/2012/01/01/symbol-error-rate-16qam-64qam-256qam/): In May 2008, we derived the theoretical symbol error rate for a general M-QAM modulation (in  Embedded.com, DSPDesignLine.com and dsplog.com) under Additive White Gaussian Noise. While re-reading that post, felt that the article is nice and warrants a re-run, using OFDM as the underlying physical layer. This post discuss the derivation of symbol error rate for a general M-QAM modulation. The companion Matlab script compares the theoretical and the simulated symbol error rate for 16QAM, 64QAM and 256QAM over OFDM in AWGN channel.
- [Newton’s method to find square root, inverse](https://dsplog.com/2011/12/25/newtons-method-square-root-inverse/): Some of us would have used Newton's method (also known as Newton-Raphson method) in some form or other. The method has quite a bit of history,  starting with the Babylonian way of finding the square root and later over centuries reaching the present recursive way of finding the solution. In this post, we will describe Newton's method and apply it to find the square root and the inverse of a number.
- [Closed form solution for linear regression](https://dsplog.com/2011/12/04/closed-form-solution-linear-regression/): In the previous post on Batch Gradient Descent and Stochastic Gradient Descent, we looked at two iterative methods for finding the parameter vector  which minimizes the square of the error between the predicted value  and the actual output  for all  values in the training set.
- [Stochastic Gradient Descent](https://dsplog.com/2011/11/15/stochastic-gradient-descent/): For curve fitting using linear regression, there exists a minor variant of Batch Gradient Descent algorithm, called Stochastic Gradient Descent.
- [Batch Gradient Descent](https://dsplog.com/2011/10/29/batch-gradient-descent/): I happened to stumble on Prof. Andrew Ng's Machine Learning classes which are available online as part of Stanford Center for Professional Development. The first lecture in the series discuss the topic of fitting parameters for a given data set using linear regression.  For understanding this concept, I chose to take data from the top 50 articles of this blog based on the pageviews in the month of September 2011.
- [Back!](https://dsplog.com/2011/10/22/back/): Am getting ready to be back posting frequently.
- [Non coherent demodulation of pi/8 D8PSK (TETRA)](https://dsplog.com/2010/07/20/non-coherent-demodulation-of-pi8-d8psk-tetra/): In TETRA specifications, one of the modulation technique used is Differential 8 Phase Shift Keying (D8PSK). We will discuss the bit error rate with non-coherent demodulation of D8PSK in Additive White Gaussian Noise (AWGN) channel.
- [IEEE 802.11ac – Very High Throughput for lower 6GHz band](https://dsplog.com/2010/04/26/80211ac-very-high-throughput-lower-6ghz/): IEEE 802.11ac Very High Throughput (for <6GHz band) is an upcoming standard which is development by IEEE standardization committee. The mandate of Task Group AC is supposed to enhance the High Throughput rates achieved by 802.11n.  As described in the document VHT below 6GHz PAR plus 5C's (802.11-08/0807r4) the group has the following objectives :
- [Non coherent demodulation of pi/4 DQPSK (TETRA)](https://dsplog.com/2010/04/12/non-coherent-demodulation-of-pi4-dqpsk-tetra/): In TETRA specifications, one of the modulation technique used is Differential Quaternary Phase Shift Keying (DQPSK). We will discuss the bit error rate with non-coherent demodulation of DQPSK in Additive White Gaussian Noise (AWGN) channel.
- [TETRA Air interface specifications – Overview](https://dsplog.com/2010/02/02/tetra-air-interface-specifications/): TETRA (TErrestrial Trunked RAdio) is a wireless specification intended to be used by government agencies, public health services, rail transport, military etc. TETRA specification comes from ETSI (European Telecommunication Standards Institute). We plan to start a post series on the building blocks of TETRA physical layer and radio specifications and this is the first post towards that step.
- [BER for BPSK in ISI channel with MMSE equalization](https://dsplog.com/2010/01/24/ber-bpsk-isi-channel-mmse-equalization/): In the past, we had discussed BER for BPSK in flat fading Rayleigh channel and BER for BPSK in a frequency selective channel using Zero Forcing Equalization. In this post, lets discuss a frequency selective channel with the use of Minimum Mean Square Error (MMSE) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.
- [Happy New Year 2010](https://dsplog.com/2009/12/31/happy-new-year-2010/): Its been a mixed year for dsplog. Some key milestones
- [BER for BPSK in ISI channel with Zero Forcing equalization](https://dsplog.com/2009/11/29/ber-bpsk-isi-channel-zero-forcing-equalization/): In the past, we had discussed BER for BPSK in flat fading Rayleigh channel. In this post, lets discuss a frequency selective channel with the use of Zero Forcing (ZF) equalization to compensate for the inter symbol interference (ISI). For simplifying the discussion, we will assume that there is no pulse shaping at the transmitter. The ISI channel is assumed to be a fixed 3 tap channel.
- [dspLog turns two! Happy Birthday!](https://dsplog.com/2009/10/21/dsplog-turns-two-happy-birthday/): As on 21st October 2007, we moved from blogspot domain at  to the self hosted domain at https://dsplog.com. Over the last two years, the blog has grown quite a bit, and am reasonably happy with the progress. Let us look back at the positives and negatives over the last year.
- [Quiz on IEEE 802.11a specifications](https://dsplog.com/2009/10/07/quiz-ieee-80211a-specifications/): The IEEE 802.11a specifications are used by many to understand a wireless communication link built using OFDM. In this post, I have put together a set of 10 multiple choice questions based on 802.11a specifications. The questions are on the building blocks in 802.11a specifications, preamble structure and so on. Upon completion of the quiz, you will be lead to a page showing the correct answers and their explanations.
- [Hamming (7,4) code with hard decision decoding](https://dsplog.com/2009/09/29/hamming-74-code-with-hard-decision-decoding/): In previous posts, we have discussed convolutional codes with Viterbi decoding (hard decision, soft decision and with finite traceback). Let us know discuss a block coding scheme where a group of information bits is mapped into coded bits. Such codes are referred to as codes. We will restrict the discussion to Hamming codes, where 4 information bits are mapped into 7 coded bits. The performance with and without coding is compared using BPSK modulation in AWGN only scenario.
- [OCW: Communication System Design](https://dsplog.com/2009/08/25/ocw-communication-system-design/): While browsing through the web for materials on the wireless communication and implementation, found this rich set of articles as part of MIT OPEN COURSEWARE program. The course is from Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
- [Matlab or C for Viterbi Decoder?](https://dsplog.com/2009/08/21/matlab-or-c-for-viterbi-decoder/): Are you bothered by speed of the speed of the simulations which you develop in Matlab/Octave? I was not bothered much, till I ran into the Viterbi decoder. If you recall, the Matlab/Octave simulation script for BER computation with hard soft decision Viterbi algorithm provided in post Viterbi with finite survivor state memory took around 10 hours to run.
- [Inter Carrier Interference (ICI) in OFDM due to frequency offset](https://dsplog.com/2009/08/08/effect-of-ici-in-ofdm/): In this post, let us evaluate the impact of frequency offset resulting in Inter Carrier Interference (ICI) while receiving an OFDM modulated symbol. We will first discuss the OFDM transmission and reception, the effect of frequency offset and later we will define the loss of orthogonality and resulting signal to noise ratio (SNR) loss due to the presence of frequency offset. The analysis is accompanied by Matlab/Octave simulation scripts.
- [Viterbi with finite survivor state memory](https://dsplog.com/2009/07/27/viterbi-with-finite-survivor-state-memory/): In the post on Viterbi decoder and soft input Viterbi decoder, we discussed a convolutional encoding scheme with rate 1/2, constraint length and having generator polynomial and having generator polynomial . If the number of uncoded bits is , then the number of coded bits at the output of the convolutional encoder is . Decoding the convolutionaly encoded bits by Viterbi algorithm consisted of the following steps.
- [Softbit for 16QAM](https://dsplog.com/2009/07/05/softbit-16qam/): In the post on Soft Input Viterbi decoder, we had discussed BPSK modulation with convolutional coding and soft input Viterbi decoding in AWGN channel. Let us know discuss the derivation of soft bits for 16QAM modulation scheme with Gray coded bit mapping. The channel is assumed to be AWGN alone.
- [Support Vibha’s Dream Mile event](https://dsplog.com/2009/06/19/donate-vibha-dream-mile/): My friend Mr. Balaji volunteers for Vibha, a non-profit  organization whose mission is to ensure that every underprivileged child attains his or her right to education, health and opportunity.
- [MSK transmitter and receiver](https://dsplog.com/2009/06/16/msk-transmitter-receiver/): In a post on Minimum Shift Keying (MSK), we had discussed that MSK uses two frequencies which are separated by and phase discontinuity is avoided in symbol boundaries. In that post, we had discussed MSK as a continuous phase transmit signal and showed that phase changes through 0, 90, 180 and 270 degrees. In this post, we will discuss MSK transmission as a variant of offset-QPSK technique. Further, we will discuss the receiver structure and show that bit error rate with coherent demodulation of MSK (using time) is equivalent to that of BPSK modulation. The channel assumed is AWGN.
- [Solved objective questions (GATE)](https://dsplog.com/2009/05/17/gate-objective-questions-solved/): Using the services of a new author 'RV', we are starting a new series of articles in the blog. Typically in India, many of the competitive examinations pertaining to Engineering (GATE, IES) and rectuitment by private and public sector companies (ISRO, BSNL, BEL, BHEL) uses examination with objective questions for the first level screening. We are hoping that this Objective Question Answer series, mostly discussing topics pertaining to Electronics and Communication Engineering, will help those preparing for those examination. Kindly do give your feedback via comment and/or email. Thanks, Krishna
- [BER with Matched Filtering](https://dsplog.com/2009/05/08/ber-with-matched-filtering/): In the post on transmit pulse shaping filter, we had discussed pulse shaping using rectangular and sinc. In this post we will discuss about optimal receiver structure when pulse shaping is used at the transmitter. The receiver structure is also called as matched filter. For the discussion, we will assume rectangular pulse shaping, the channel is AWGN only and the modulation is BPSK.
- [Six equalizers for V-BLAST](https://dsplog.com/2009/04/21/six-equalizers-for-v-blast/): In the past, we had discussed several posts on two transmit two receive MIMO communication, where the transmission was based on V-BLAST. The details about V-BLAST can be read from the landmark paper V-BLAST: An architeture for realizing very high data rates over the rich scattering wireless channel - P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Transmit beamforming](https://dsplog.com/2009/04/13/transmit-beamforming/): In this post lets discuss a closed-loop transmit diversity scheme, where the transmitter has the knowledge of the channel. As there is a feedback path required from the receiver, to communicate the channel seen by the receiver to the transmitter, the scheme is called closed-loop transmit diversity scheme. Recall that the transmit diversity using Space Time Coding (Alamouti STBC) does not require the knowledge of the channel. In this post, we will restrict our discussion to a 2 transmit, 1 receive case. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Milestone 1000+ subscribers 1100+ comments](https://dsplog.com/2009/03/28/milestone-subscribers-comments/): Those who are regular visitors to dsplog.com might have noticed the small FeedBurner chicklet on the side showing subscriber count showing 1000+ subscribers. Its a nice milestone to reach, one that looked so distant when I wrote the first post stating the objective of this blog on 26th February 2007. We now have around 86 articles with 1100+ comments.
- [Alamouti STBC with 2 receive antenna](https://dsplog.com/2009/03/15/alamouti-stbc-2-receive-antenna/): In the past, we had discussed two transmit, one receive antenna Alamouti Space Time Block Coding (STBC) scheme. In this post, lets us discuss the impact of having two antennas at the receiver. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [IQ imbalance in transmitter](https://dsplog.com/2009/03/08/iq-imbalance-in-transmitter/): Typical communication systems use I-Q modulation and we had discussed the need for I-Q modulation in the past. In this post, let us understand I-Q imbalance and its effect on transmit signal.
- [Approximate Vector Magnitude Computation](https://dsplog.com/2009/02/08/approximate-vector-magnitude-computation/): In this post, let us discuss a simple implementation friendly scheme for computing the absolute value of a complex number . The technique called (alpha Max + beta Min) algorithm is discussed in Chapter 13.2 of Understanding Digital Signal Processing, Richard Lyons and is also available online at Digital Signal Processing Tricks - High-speed vector magnitude approximation
- [Derivation of BPSK BER in Rayleigh channel](https://dsplog.com/2009/01/22/derivation-ber-rayleigh-channel/): This is a guest post by Jose Antonio Urigüen who is an Electrical and Electronic Engineer currently studying an MSc in Communications and Signal Processing at Imperial College in London. This guest post has been created due to his own curiosity when reviewing some concepts of BER for BPSK in Rayleigh channnel published in the dsplog.com
- [Soft Input Viterbi decoder](https://dsplog.com/2009/01/14/soft-viterbi/): In two previous posts, we have discussed Convolutional Coding and the associated hard decision Viterbi decoding. In this post lets extent Viterbi decoding algorithm to soft input decision scheme. The modulation used is BPSK and the channel is assumed to be AWGN alone.
- [Migration to Who’s Who theme](https://dsplog.com/2009/01/13/whos-who-theme/): After a gap of around 7 months, we decided to migrate to a new template (recall, last time we changed was in May 2008). The template which we decided to use  is designed by Elegant Themes. They have a wide array of themes and I chose to use the Who's Who theme.
- [Viterbi decoder](https://dsplog.com/2009/01/04/viterbi/): Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the communication. Lets discuss a simple binary convolutional coding scheme at the transmitter and the associated Viterbi (maximum likelihood) decoding scheme at the receiver.
- [Convolutional code](https://dsplog.com/2009/01/04/convolutional-code/): Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the communication. In this article, lets discuss a simple binary convolutional coding scheme at the transmitter and the associated Viterbi (maximum likelihood) decoding scheme at the receiver.
- [Books](https://dsplog.com/2008/12/24/books/): Happy holidays! :)
- [MIMO with ML equalization](https://dsplog.com/2008/12/14/mimo-ml-equalization/): We have discussed quite a few receiver structures for a 2x2 MIMO channel namely,
- [Join dspLog at Google FriendConnect](https://dsplog.com/2008/12/10/dsplog-google-friendconnect/): We have installed Google FriendConnect on dspLog.com. With Google Friend Connect, you can:
- [MIMO with MMSE SIC and optimal ordering](https://dsplog.com/2008/12/06/mimo-with-mmse-sic-and-optimal-ordering/): This post attempts to build further on the MIMO equalization schemes which we have discussed -
- [MIMO with ZF SIC and optimal ordering](https://dsplog.com/2008/11/29/mimo-zf-sic-optimal-order/): In previous posts, we had discussed equalization of a 2x2 MIMO channel with Zero Forcing (ZF) equalization and later, Zero Forcing equalization with successive interference cancellation (ZF-SIC). In this post, we will explore a variant of ZF-SIC called Zero Forcing Successive Interference Cancellation with optimal ordering. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Linear to log conversion](https://dsplog.com/2008/11/20/linear-to-log-conversion/): In signal processing blocks like power estimation used in digital communication, it may be required to represent the estimate in log scale. This post explains a simple linear to log conversion scheme proposed in the DSP Guru column on DSP Trick: Quick-and-Dirty Logarithms. The scheme makes implementation of a linear to log conversion simple and small in a digital hardware like FPGA.
- [MIMO with Zero Forcing Successive Interference Cancellation equalizer](https://dsplog.com/2008/11/09/mimo-zero-forcing-successive-interference-cancellation/): The post on MIMO with Zero Forcing equalizer discussed a probable way of equalizing a 2x2 MIMO channel. The simulated results with the 2x2 MIMO system with zero forcing equalizer showed matching results as obtained in for a 1x1 system for BPSK modulation in Rayleigh channel. In this post, we will try to improve the bit error rate performance by trying out Successive Interference Cancellation (SIC). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [MIMO with MMSE equalizer](https://dsplog.com/2008/11/02/mimo-mmse-equalizer/): In a previous post, we had discussed a 2x2 MIMO transmission using BPSK modulation in Rayleigh channel with a Zero Forcing equalizer. The simulated results with the 2x2 MIMO system  with zero forcing equalizer showed matching results as obtained in for a 1x1 system for BPSK modulation in Rayleigh channel. In this post, we will discuss a different equalization approach called Minimum Mean Square Error (MMSE) equalization. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Happy Birthday – dspLog](https://dsplog.com/2008/10/24/happy-birthday-dsplog/): An important milestone for the dspLog happened on Oct 21st 2008. On this day last year, the blog migrated from the Blogger platform to the independently hosted platform at www.dsplog.com ! Belated birthday wishes for the blog!!! :)
- [MIMO with Zero Forcing equalizer](https://dsplog.com/2008/10/24/mimo-zero-forcing/): We had discussed three Single Input Multiple Output (SIMO also known as receive diversity) schemes - Selection combining, Equal Gain Combining, Maximal Ratio Combining and a Multiple Input Single Output (MISO, also known as transmit diversity) scheme - Alamouti 2x1 STBC. Let us now discuss the case where there a multiple transmit antennas and multiple receive antennas resulting in the formation of a Multiple Input Multiple Output (MIMO) channel. In this post, we will restrict our discussion to a 2 transmit 2 receive antenna case (resulting in a 2x2 MIMO channel). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Alamouti STBC](https://dsplog.com/2008/10/16/alamouti-stbc/): In the recent past, we have discussed three receive diversity schemes - Selection combining, Equal Gain Combining and Maximal Ratio Combining. All the three approaches used the antenna array at the receiver to improve the demodulation performance, albeit with different levels of complexity. Time to move on to a transmit diversity scheme where the information is spread across multiple antennas at the transmitter. In this post, lets discuss a popular transmit diversity scheme called Alamouti Space Time Block Coding (STBC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Download free e-book on error probability in AWGN](https://dsplog.com/2008/10/01/download-free-e-book/): We have quite a few articles discussing bit and symbol error rates for popular digital modulation schemes in Additive White Gaussian Noise (AWGN) channel. This post summarizes the articles discussing the theoretical and simulated error rates for the digital modulation schemes like BPSK, QPSK, 4-PAM, 16PSK and 16QAM. Further, Bit Error Rate with Gray coded mapping, bit error rate for BPSK over OFDM are also discussed.
- [Maximal Ratio Combining (MRC)](https://dsplog.com/2008/09/28/maximal-ratio-combining/): This is the third post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem - how do we use 'effectively' the information from all the antennas to demodulate the data. In the previous posts, we discussed selection diversity and equal gain combining (EGC).
- [Equal Gain Combining (EGC)](https://dsplog.com/2008/09/19/equal-gain-combining/): This is the second post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem - how do we use 'effectively' the information from all the antennas to demodulate the data. In the previous post, we discussed selection diversity. In this post, we will discuss equal gain combining (EGC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [Selection Diversity](https://dsplog.com/2008/09/06/receiver-diversity-selection-diversity/): This is the first post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem - how do we use 'effectively' the information from all the antennas to demodulate the data. There are multiple ways to approach the problem. The three typical approaches to be discussed are - selection diversity, equal gain combining and maximal ratio combining. In this post we will discuss selection diversity. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.
- [BER for BPSK in OFDM with Rayleigh multipath channel](https://dsplog.com/2008/08/26/ofdm-rayleigh-channel-ber-bpsk/): Mr. Lealem Tamirat, in a comment on BER for BPSK in Rayleigh channel, wondered about the performance of an OFDM modulated system in a frequency selective Rayeligh fading channel. My response was that,
- [Receive diversity in AWGN](https://dsplog.com/2008/08/19/receive-diversity-in-awgn/): Some among you will be aware that in a wireless link having multiple antenna's at the receiver (aka receive diversity) improves the bit error rate (BER) performance. In this post, let us try to understand the BER improvement with receive diversity. And, since we are just getting started, let us limit ourselves to additive white Gaussian noise (AWGN) channel (i.e assume that the channel gains are unity).
- [BER for BPSK in Rayleigh channel](https://dsplog.com/2008/08/10/ber-bpsk-rayleigh-channel/): Long back in time we discussed the BER (bit error rate) for BPSK modulation in a simple AWGN channel (time stamps states August 2007). Almost an year back! It high time we discuss the BER for BPSK in a Rayleigh multipath channel.
- [Negative Frequency](https://dsplog.com/2008/08/08/negative-frequency/): Last week, I received an email from Mr. Kishore. He was wondering about the physical significance of negative frequency. Does negative frequency really exist?
- [Summary – feedback on [dspLog], July 2008](https://dsplog.com/2008/08/03/feedback-summary-dsplog/): On July30th, 2008 I had sent a request for feedback to 93 subscribers who have opted to receive articles over email. As on 3rd August, I received the response from around 8 persons. Not bad, around 8.5% response. Thanks a lot for the feedback. I will summarize the response from the group and note down the action items on me.
- [OT: Happy Schools Blog](https://dsplog.com/2008/07/30/happy-schools-blog/): Mr. Raghuram contacted me and informed about Happy Schools Blog.
- [Chi Square Random Variable](https://dsplog.com/2008/07/28/chi-square-random-variable/): While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss chi square random variable prior.
- [ICCBN 2008, July 17-20  2008, IISc, Bangalore](https://dsplog.com/2008/07/18/iccbn-2008-july-2008-iisc-bangalore/): Advanced Computing and Communication Society (ACS) of India is organizing ICCBN 2008 conference (International Conference on Communication, Convergence, and Broadband Networking) from July 17th to 20th 2008 at National Science Seminar Complex at Indian Institute of Science (IISc), Bangalore.
- [Deriving PDF of Rayleigh random variable](https://dsplog.com/2008/07/17/derive-pdf-rayleigh-random-variable/): In the post on Rayleigh channel model, we stated that a circularly symmetric random variable is of the form , where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables. The magnitude which has the probability density,
- [Rayleigh multipath channel model](https://dsplog.com/2008/07/14/rayleigh-multipath-channel/): The article gives a quick overview of a simple statistical multipath channel model called Rayleigh fading channel model.
- [Comparing BPSK, QPSK, 4PAM, 16QAM, 16PSK, 64QAM and 32PSK](https://dsplog.com/2008/07/08/compare-bpsk-qpsk-4pam-16qam-16psk-64qam-32psk/): I have written another article in DSPDesginLine.com. This article can be treated as the third post in the series aimed at understanding Shannon's capacity equation.
- [Trying out PAPR reduction for OFDM by multiplication with j](https://dsplog.com/2008/07/02/ofdm-papr-reduction-multiplication-by-j/): In this post, we will explore a probable way of reducing PAPR (peak to average power ratio) in OFDM by changing the phase of some of the subcarriers. This is in response to the comment to post on Peak to Average power ratio for OFDM, where Mr. Elibom suggested to reduce the PAPR by cyclically rotate some of the subcarriers and using.
- [Bounds on Communication based on Shannon’s capacity](https://dsplog.com/2008/06/18/bounds-on-communication-shannon-capacity/): This is the second post in the series aimed at developing a better understanding of Shannon's capacity equation. In this post let us discuss the bounds on communication given the signal power and bandwidth constraint. Further, the following writeup is based on Section 12.6 from Fundamentals of Communication Systems by John G. Proakis, Masoud Salehi
- [Understanding Shannon’s capacity equation](https://dsplog.com/2008/06/15/shannon-gaussian-channel-capacity-equation/): Let us try to understand the formula for Channel Capacity with an Average Power Limitation, described in Section 25 of the landmark paper A Mathematical Theory for Communication, by Mr. Claude Shannon.
- [BPSK BER with OFDM modulation](https://dsplog.com/2008/06/10/ofdm-bpsk-bit-error/): Oflate, I am getting frequent requests for bit error rate simulations using OFDM (Orthogonal Frequency Division Multiplexing) modulation. In this post, we will discuss a simple OFDM transmitter and receiver, find the relation between Eb/No (Bit to Noise ratio) and Es/No (Signal to Noise ratio) and compute the bit error rate with BPSK.
- [16QAM Bit Error Rate (BER) with Gray mapping](https://dsplog.com/2008/06/05/16qam-bit-error-gray-mapping/): Let us derive the theoretical 16QAM bit error rate (BER) with Gray coded constellation mapping in additive white Gaussian noise conditions. Further, the Matlab/Octave simulation script can be used to confirm that the simulation is in good agreement with theory.
- [Binary to Gray code for 16QAM](https://dsplog.com/2008/06/01/binary-to-gray-code-for-16qam/): In the previous post on Binary to Gray code conversion for PSK, I had claimed that "for a general M-QAM modulation the binary to Gray code conversion is bit more complicated". However following a closer look, I realize that this is not so complicated. :)
- [Update: Correction in Matlab code for raised cosine filter](https://dsplog.com/2008/05/25/update-correction-in-matlab-code-for-raised-cosine-filter/): Thanks to the keen observation by Mr. Phan Minh Hoang, I was notified that the Matlab/Octave scripts provided along with the topic raised cosine filtering was not behaving properly.
- [Article in DSPDesignLine.com: M-QAM symbol error](https://dsplog.com/2008/05/24/article-in-dspdesignlinecom-m-qam-symbol-error/): Its been a nice week for me, wherein I guest posted an article in DSPDesignLine.com. :)
- [Bit error rate for 16PSK modulation using Gray mapping](https://dsplog.com/2008/05/18/bit-error-rate-for-16psk-modulation-using-gray-mapping/): In this post, let us derive the theoretical bit error probability for 16PSK modulation using Gray coded mapping. For deriving the equation, we will refer material from the following posts:
- [Gray code to Binary conversion for PSK and PAM](https://dsplog.com/2008/05/12/gray-code-to-binary-conversion-for-psk-pam/): Given that we have discussed Binary to Gray code conversion, let us discuss the Gray to BInary conversion.
- [Binary to Gray code conversion for PSK and PAM](https://dsplog.com/2008/05/11/binary-to-gray-code-conversion-psk-pam/): In this post, let us try to understand Gray codes and their usage in digital communication. Quoting from Wiki entry on Gray code ,
- [OT: Migration to a Deep Blue template](https://dsplog.com/2008/05/04/dsplog-to-deep-blue-template/): After almost 6 months with the Smashing Theme template, its time for a change. Recently I stumbled upon the Deep Blue template from DailyBlogTips. I liked the clean blue-green-white combination and and felt it might be a good fit for www.dsplog.com. Hope you agree.
- [Eye diagram with raised cosine filtering](https://dsplog.com/2008/05/01/eye-diagram-plot-matlab-raised-cosine-filter/): We have discussed about probable transmit pulse shaping filter and have observed that raised cosine filtering filtering allows a simpler implementation, albeit at the cost of increased bandwidth. Let us know understand the eye diagram, which is a useful graphical tool to quantify the degradation of the signal due to filtering.
- [MCDES 2008 at Indian Institute of Science, Bangalore](https://dsplog.com/2008/04/23/mcdes-2008-iisc-bangalore-may-2008/): Thanks to Prof. K V S Hari, I was informed that the centenary year for Indian Institute of Science (IISc), Bangalore starts on May 27th 2008.
- [Raised cosine filter for transmit pulse shaping](https://dsplog.com/2008/04/22/raised-cosine-filter-for-transmit-pulse-shaping/): In the previous post on transmit filtering using Nyquist pulse, we had briefly learned that the information symbol with a symbol period can be transmitted without inter symbol interference (ISI) by using Nyquist pulse,
- [OT: Prof. Randy Pausch’s lecture in Oprah show](https://dsplog.com/2008/04/19/prof-randy-pausch-lecture-oprah-show/): In the past, I have wondered about discussing personal thoughts in this blog. The answer in my mind was NO and I ve been focusing only on technical topics till date. However, there is a change of mind, thanks to my friend Manoj.
- [Transmit pulse shaping filter – rectangular and sinc (Nyquist)](https://dsplog.com/2008/04/14/transmit-pulse-shape-nyquist-sinc-rectangular/): In the previous post on I-Q modulator and de-modulator, we had briefly mentioned that the a baseband PAM transmission can be modelled as
- [IQ modulation and demodulation](https://dsplog.com/2008/04/05/need-for-i-q-modulator-and-demodulator/): Post describes about the need for I-Q modulation by comparing the spectral efficiency of passband PAM and passband QAM.
- [Comparing 16PSK vs 16QAM for symbol error rate](https://dsplog.com/2008/03/29/comparing-16psk-vs-16qam-for-symbol-error-rate/): In two previous posts, we have derived theoretical symbol error rate for 16-QAM and 16-PSK modulation schemes. The links are:
- [Blog on DSP](https://dsplog.com/2008/03/25/blog-on-dsp/): I happened to visit 'The Digital Signal Processing Blog', maintained by Mr. Andres Kwasinski, Ph. D.
- [Symbol Error Rate for 16PSK](https://dsplog.com/2008/03/18/symbol-error-rate-for-16psk/): In this post, let us try to derive the symbol error rate for 16-PSK (16-Phase Shift Keying) modulation.
- [Frequency offset estimation using 802.11a short preamble](https://dsplog.com/2008/03/03/frequency-offset-estimation-using-80211a-short-preamble/): From the previous post on OFDM (here), we have understood that an OFDM waveform is made of sum of multiple sinusoidals (also called subcarriers) each modulated independently. In this post, let us try to understand the estimation of frequency offset in a typical OFDM receiver (using the short preamble specified per IEEE 802.11a specification as a reference).
- [Peak to Average Power Ratio for OFDM](https://dsplog.com/2008/02/24/peak-to-average-power-ratio-for-ofdm/): Let us try to understand peak to average power ratio (PAPR) and its typical value in an OFDM system specified per IEEE 802.11a specifications.
- [Cylcic prefix in Orthogonal Frequency Division Multiplexing](https://dsplog.com/2008/02/17/cylcic-prefix-in-orthogonal-frequency-division-multiplexing/): In a previous post (here), we discussed in brief, Orthogonal Frequency Division Multiplexing (OFDM) transmission. Let us know probe bit more into the motivation of cyclic prefix (aka guard interval) associated with each OFDM symbol.
- [Understanding an OFDM transmission](https://dsplog.com/2008/02/03/understanding-an-ofdm-transmission/): Let us try to understand simulation of a typical Orthogonal Frequency Division Multiplexing (OFDM) transmission defined per IEEE 802.11a specification.
- [Simulating Minimum Shift Keying Transmitter](https://dsplog.com/2008/01/19/simulating-minimum-shift-keying-transmitter/): Minimum shift keying (MSK) is an important concept to learn in digital communications. It is a form of continuous phase frequency shift keying . In minimum phase shift keying, two key concepts are used.
- [Minimum frequency spacing for having orthogonal sinusoidals](https://dsplog.com/2007/12/31/minimum-frequency-spacing-for-having-orthogonal-sinusoidals/): In this post, the objective is to figure out the minimum separation between two sinusoidals having frequencies , of duration each to be orthogonal. Let the phase difference between the sinusoidals is where can take any value from to (Refer Example 4.3 ).
- [Using CORDIC for phase and magnitude computation](https://dsplog.com/2007/12/16/using-cordic-for-phase-and-magnitude-computation/): In a previous post (here), we looked at using CORDIC (Co-ordinate Rotation by DIgital Computer) for understanding how a complex number can be rotated by an angle without using actual multipliers. Let us know try to understand how we can use CORDIC for finding the phase and magnitude of a complex number.
- [Symbol Error Rate (SER) for 16-QAM](https://dsplog.com/2007/12/09/symbol-error-rate-for-16-qam/): Given that we have went over the symbol error probability for 4-PAM and symbol error rate for 4-QAM , let us extend the understanding to find the symbol error probability for 16-QAM (16 Quadrature Amplitude Modulation). Consider a typical 16-QAM modulation scheme where the alphabets (Refer example 5-37 in ).
- [Digital implementation of RC low pass filter](https://dsplog.com/2007/12/02/digital-implementation-of-rc-low-pass-filter/): Thanks to the nice article from Xilinx TechXclusives , let us try to understand the probable digital implementation of resistor-capacitor based low pass filter. Consider a simple RC filter shown in the figure below. Assuming that there is no load across the capacitor, the capacitor charges and discharges through the resistor path.
- [Symbol Error Rate (SER) for QPSK (4-QAM) modulation](https://dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/): Given that we have discussed symbol error rate probability for a 4-PAM modulation, let us know focus on finding the symbol error probability for a QPSK (4-QAM) modulation scheme.
- [Symbol Error Rate (SER) for 4-PAM](https://dsplog.com/2007/10/07/symbol-error-rate-for-pam/): Following discussion of bit error rate (BER) for BPSK and bit error rate for FSK, it may be interesting to move on to discuss a higher order constellation such as Pulse Amplitude Modulation (PAM).
- [Coherent demodulation of DBPSK](https://dsplog.com/2007/09/30/coherent-demodulation-of-dbpsk/): In a previous post, we discussed about a probable first order digital PLL for tracking constant phase offset. The assumption was that as the phase offset is small and the bits gets decoded correctly, the phase difference between the ideal and actual constellation gives the initial value of phase. However, in typical scenarios it may be possible that the above assumption may not be valid, resulting in phase ambiguity.
- [Scaling factor in QAM](https://dsplog.com/2007/09/23/scaling-factor-in-qam/): When QAM (Quadrature Amplitude Modulation) is used, typically one may find a scaling factor associated with the constellation mapping operation. It may be reasonably obvious that this scaling factor is for normalizing the average energy to one.
- [Bit Error Rate (BER) for frequency shift keying with coherent demodulation](https://dsplog.com/2007/08/30/bit-error-rate-for-frequency-shift-keying-with-coherent-demodulation/): Following the request by Siti Naimah, this post discuss the bit error probability for coherent demodulation of binary Frequency Shift Keying (BFSK) along with a small Matlab code snippet.
- [CORDIC for phase rotation](https://dsplog.com/2007/08/19/cordic-for-phase-rotation/): My understanding of the CORDIC (Co-ordinate Rotation by DIgital Computer) thanks to the nice article in .
- [Bit Error Rate (BER) for BPSK modulation](https://dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/): In this post, we will derive the theoretical equation for bit error rate (BER) with Binary Phase Shift Keying (BPSK) modulation scheme in Additive White Gaussian Noise (AWGN) channel. The BER results obtained using Matlab/Octave & Python simulation scripts show good agreement with the derived theoretical results.
- [Straight line fit using least squares estimate](https://dsplog.com/2007/07/15/straight-line-fit-using-least-squares-estimate/): Two points suffice for drawing a straight line. However we may be presented with a set of data points (more than two?) presumably forming a straight line. How can one use the available set of data points to draw a straight line?
- [Example of Cascaded Integrator Comb filter in Matlab](https://dsplog.com/2007/07/01/example-of-cascaded-integrator-comb-filter-in-matlab/): Let me briefly share my understanding on the cascaded integrator comb (CIC) filter, thanks to the nice article. For understanding the cascaded integrator comb (CIC) filter, firstly let us understand the moving average filter, which is accumulation latest samples of an input sequence .
- [Interpreting the output of fft() operation in Matlab](https://dsplog.com/2007/06/17/interpreting-the-output-of-fft-operation-in-matlab/): It might be interesting to interpret the output of the fft() function in Matlab. Consider the following simple examples.
- [First order digital PLL for tracking constant phase offset](https://dsplog.com/2007/06/10/first-order-digital-pll-for-tracking-constant-phase-offset/): Considering a typical scenario where there might exist a small phase offset between local oscillator between the transmitter and receiver.
- [Polyphase filters for interpolation](https://dsplog.com/2007/05/12/polyphase-filters-for-interpolation/): In typical digital signal processing applications, there arises need to increase the sampling frequency of a signal sequence, where the higher sampling frequency is an integer multiple of the original sampling frequency i.e for a signal sequence with a sampling frequency , change the sampling frequency to , where is an integer.
- [Using Toeplitz matrices in MATLAB](https://dsplog.com/2007/04/21/using-toeplitz-matrices-in-matlab/): The definition of Toeplitz matrix from is:
- [2nd order sigma delta modulator](https://dsplog.com/2007/04/08/2nd-order-sigma-delta-modulator/): In a previous post, the variance of the in-band quantization noise for a first order sigma delta modulator was derived. Taking it one step furhter, let us find the variance of the quantization noise filtered by a second order filter.
- [Sigma delta modulation](https://dsplog.com/2007/04/03/sigma-delta-modulation/): In an earlier post, it was mentioned that delta modulator without the quantizer is identical to convolving an input sequence with . Let us first try to validate that thought using a small MATLAB example and using the delta modulator circuit shown in Figure 9.13a of DSP-Proakis .
- [Zero-order hold and first-order hold based interpolation](https://dsplog.com/2007/03/25/zero-order-hold-and-first-order-hold-based-interpolation/): In problem 9.14 of DSP-Proakis, the objective is to analyze the effect of zero-order interpolation and first-order interpolation to double the number of samples in the sinusoidal
- [Signal to quantization noise in quantized sinusoidal](https://dsplog.com/2007/03/19/signal-to-quantization-noise-in-quantized-sinusoidal/): In problem 4.37 of DSP-Proakis , the task is to analyze the total harmonic distortion in quantized sinusoidal, where .
- [Harmonic distortion in digital sinusoidal generators](https://dsplog.com/2007/03/18/harmonic-distortion-in-digital-sinusoidal-generators/): As the signal is real valued, the power at other frequency components apart from the desired frequency component is .
- [LaTex on blogger not working?](https://dsplog.com/2007/03/16/latex-on-blogger-not-working/): I noticed that the scripts provided from http://wolverinex02.googlepages.com/emoticonsforblogger2 for writing equations on blogger stopped working.
- [Modifying transmit signal](https://dsplog.com/2007/03/11/modifying-transmit-signal/): In DSP-Proakis, I found the problem 2.56 interesting as it provides insight into delta modulation. The task is to show that,
- [Trying out LaTeX on Blogspot](https://dsplog.com/2007/03/07/trying-out-latex-on-blogspot/): Quick check of on Blogspot, thanks to the information provided here.
- [Objective](https://dsplog.com/2007/02/26/objective/): The blog attempts to provide and collect information on digital signal processing algorithms and implementation.