Signal Processing Algorithms

Why does signal processing math feel hard? It’s not the math. It’s how we learn it. We memorise symbols. We rarely build systems. I was that telecom engineer who froze at complex numbers and stared at Fourier transforms like a foreign script. It clicked only when I taught five students and forced myself to tie every idea to a real signal and a small build. Here’s the 7‑day fix I use with the 1:1:1 method: one concept, one real example, one tiny project. Day 1 — Complex numbers Concept: phasors and Euler’s formula. Example: ECG lead II. Build: compute magnitude/phase of a synthetic ECG in Python and plot. Day 2 — Sampling Concept: Nyquist and aliasing. Example: audio clip at 8 kHz vs 44.1 kHz. Build: resample and hear aliasing artefacts. Day 3 — Discrete Fourier Transform Concept: frequency bins and resolution. Example: EEG alpha (8–12 Hz). Build: FFT on a 10‑second EEG segment, mark the alpha peak. Day 4 — Windows Concept: leakage and window choice. Example: Hamming vs rectangular on a two‑tone signal. Build: compare spectra; note sidelobes. Day 5 — FIR/IIR filters Concept: magnitude/phase, stability, delay. Example: EMG powerline hum at 50 Hz. Build: design a notch and a band‑pass; show before/after. Day 6 — Convolution Concept: LTI view of filtering. Example: moving average on PPG to smooth motion noise. Build: implement conv1d and compare with library output. Day 7 — End‑to‑end mini project Concept: tie it all together. Example: detect resting alpha from EEG or remove 50 Hz hum from EMG. Build: clean the signal, extract one feature, and output a simple decision. Keep it small. One plot, one metric, one decision. That’s enough to turn a formula into a skill. This is how we can go from “FFT makes no sense” to shipping small builds and land roles within six months. If you want my simple roadmap for learning math for signal processing, its below, no need to say “roadmap” in comments. Found this helpful? If yes, 👍 Like and ♻️ reshare with your friends. _________ Join 1.7K+ learners in transforming how we learn follow Shankar J. for more.

Pulse extraction the basics To extract a pulse from a carrier signal, you can use a technique called pulse detection or pulse demodulation. Here's a simple algorithm that outlines the process: 1. Obtain the carrier signal: Start by capturing or acquiring the carrier signal that contains the pulses you want to extract. 2. Preprocessing: Filter the carrier signal to remove any noise or unwanted frequencies that may interfere with pulse detection. Common techniques include low-pass filtering or band-pass filtering. 3. Envelope detection: Use an envelope detection technique to extract the amplitude variations of the carrier signal. One common method is to rectify the signal by removing the negative half-cycles, resulting in a unipolar waveform. 4. Thresholding: Set a threshold value to distinguish between the presence and absence of pulses. This threshold can be determined based on the characteristics of the carrier signal. Any amplitude above the threshold is considered to be a pulse. 5. Pulse detection: Scan through the envelope-detected signal and identify the instances where the signal crosses the threshold. Mark these instances as pulse occurrences. 6. Pulse extraction: Once the pulse occurrences are identified, you can extract the pulse waveform by locating the start and end points of each pulse. This can be done by finding the rising and falling edges of the pulse signals. 7. Post-processing: Perform any necessary post-processing on the extracted pulses, such as noise removal or signal conditioning, to enhance the quality or accuracy of the extracted pulse signals. It's important to note that the specific implementation details and parameters of the algorithm may vary depending on the characteristics of the carrier signal and the specific application.

In many digital signal processing (DSP) applications within FPGAs, calculating the magnitude of a vector is essential. This is especially true for tasks such as automatic gain control (AGC) loops, demodulation, and equalization, to name just a few. Typically, this operation is implemented using CORDIC algorithms or other iterative methods. However, there is a faster and more efficient alternative: the 𝗔𝗹𝗽𝗵𝗮 𝗠𝗮𝘅 𝗣𝗹𝘂𝘀 𝗕𝗲𝘁𝗮 𝗠𝗶𝗻 algorithm. This algorithm provides a quick approximation of the vector’s magnitude with minimal error. The error depends on the values of the alpha and beta coefficients, and with optimal coefficients, the estimation error is typically less than 0.5 dB. In some variations, the error can even be reduced to as low as 0.1 dB. What sets this algorithm apart is its simplicity compared to traditional methods. It can be further optimized by selecting coefficients as powers of two or using the canonical signed-digit (CSD) representation. The attached figure shows a MathWorks MATLAB simulation of the Alpha Max Plus Beta Min algorithm applied to AM signal demodulation (envelope detection), along with error variations for different coefficient values. If you're interested in exploring this algorithm further, I recommend checking out the following link: https://lnkd.in/dZvVJSYd #DSP #FPGA #VHDL #Matlab

The Magic Behind MUSIC Multiple Signal Classification (MUSIC) is a smart technique used to detect the direction of incoming signals. While the name may sound artistic, it is actually a mathematical method that helps systems like 5G and WiFi determine where a signal is coming from. It is widely used in communication, radar, sonar, and even in medical imaging [1]. What Does MUSIC Do? Imagine placing several antennas in a straight line. When a signal from a phone or transmitter reaches these antennas, it arrives at each one with slight differences in timing and phase. MUSIC analyzes these differences to determine the direction the signal came from, a process known as Direction of Arrival (DoA) estimation [2]. To achieve this, MUSIC uses what is known as an antenna array, which is a group of antennas working together. When signals arrive at the array, the algorithm forms a covariance matrix based on the received data. This matrix is then divided into two parts. One part contains the useful signal information, called the signal subspace, and the other part contains background noise, called the noise subspace [3]. By scanning across all possible directions, MUSIC calculates a "spectrum" and looks for peaks. These peaks show the directions of the incoming signals. The key equation is: P(θ) = 1 / ||a(θ)* En||², where: -P(θ) is the spatial spectrum at angle θ -a(θ) is the array steering vector for angle θ -En is the noise subspace matrix -* means Hermitian transpose (complex conjugate transpose) -||...|| is the norm (size of the vector) When a(θ) aligns poorly with En, the value of P(θ) becomes large, which indicates the true angle of arrival [3]. Imagine you have two puzzles: one represents random noise direction, and the other represents real signal direction. You take a puzzle piece, representing a potential signal direction (a(θ)), and try to fit it into the noise puzzle (En). Most pieces don’t fit well, but occasionally, a piece doesn’t fit at all, it has zero overlap. That perfect mismatch tells you the piece actually belongs to the real signal puzzle. In MUSIC, this total misfit with the noise space leads to a sharp peak in the spectrum, revealing the true direction of the signal. MUSIC enables 5G networks to pinpoint user locations precisely, boosting data rates and cutting interference [4]. It helps base stations focus signals directly at devices instead of broadcasting broadly. In WiFi, MUSIC has been applied for indoor tracking and even through-wall sensing by analyzing signal reflections [5,6]. Recently, MUSIC has become important in joint communication and sensing (JCAS) systems, especially in automotive radar and 6G research. MUSIC allows systems to identify object directions while maintaining reliable data transmission [7]. Takeaway MUSIC is a clever and efficient method that helps modern wireless systems detect where signals come from. It enables 5G and WiFi to perform better by accurately finding user directions.