A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters
This article discusses the behavior, mathematics, and implementation of cascaded integrator-comb filters.
Understanding and Implementing the Sliding DFT
Introduction In many applications the detection or processing of signals in the frequency domain offers an advantage over performing the same task in the time-domain. Sometimes the advantage is just a simpler or more conceptually...
How the Cooley-Tukey FFT Algorithm Works | Part 2 - Divide & Conquer
The Fast Fourier Transform revolutionized the Discrete Fourier Transform by making it much more efficient. In part 1, we saw that if you run the DFT on a power-of-2 number of samples, the calculations of different groups of samples repeat themselves at different frequencies. By leveraging the repeating patterns of sine and cosine values, the algorithm enables us to calculate the full DFT more efficiently. However, the calculations of certain groups of samples repeat more often than others. In this article, we’re going to explore how the divide-and-conquer method prepares the ground for the next stage of the algorithm by grouping the samples into specially ordered pairs.
Spectral Flipping Around Signal Center Frequency
Most of us are familiar with the process of flipping the spectrum (spectral inversion) of a real signal by multiplying that signal's time samples by (-1)n. In that process the center of spectral rotation is fs/4, where fs is the signal's sample...
The First-Order IIR Filter -- More than Meets the Eye
While we might be inclined to disdain the simple first-order infinite impulse response (IIR) filter, it is not so simple that we can’t learn something from it. Studying it can teach DSP math skills, and it is a very useful filter in its own right. In this article, we’ll examine the time response of the filter, compare the first-order IIR filter to the FIR moving average filter, use it to smooth a noisy signal, compute the functional form of the impulse response, and find the frequency response.
How the Cooley-Tukey FFT Algorithm Works | Part 4 - Twiddle Factors
The beauty of the FFT algorithm is that it does the same thing over and over again. It treats every stage of the calculation in exactly the same way. However, this. “one-size-fits-all” approach, although elegant and simple, causes a problem. It misaligns samples and introduces phase distortions during each stage of the algorithm. To overcome this, we need Twiddle Factors, little phase correction factors that push things back into their correct positions before continuing onto the next stage.
Two Easy Ways To Test Multistage CIC Decimation Filters
This article presents two very easy ways to test the performance of multistage cascaded integrator-comb (CIC) decimation filters. Anyone implementing CIC filters should take note of the following proposed CIC filter test methods.
Goertzel Algorithm for a Non-integer Frequency Index
If you've read about the Goertzel algorithm, you know it's typically presented as an efficient way to compute an individual kth bin result of an N-point discrete Fourier transform (DFT). The integer-valued frequency index k is in the range of...
Understanding and Relating Eb/No, SNR, and other Power Efficiency Metrics
Introduction Evaluating the performance of communication systems, and wireless systems in particular, usually involves quantifying some performance metric as a function of Signal-to-Noise-Ratio (SNR) or some similar measurement. Many systems...
How the Cooley-Tukey FFT Algorithm Works | Part 1 - Repeating Calculations
The Fourier Transform is a powerful tool, used in many technologies, from audio processing to wireless communication. However, calculating the FT can be computationally expensive. The Cooley-Tukey Fast Fourier Transform (FFT) algorithm provides a significant speedup. It exploits the repetitive nature of calculations within the Discrete Fourier Transform (DFT), the mathematical foundation of the FT. By recognizing patterns in the DFT calculations and reusing intermediate results, the FFT vastly reduces the number of operations required. In this series of articles, we will look at how the Cooley-Tukey FFT algorithm works.
Fixed-Point Simulation in GNU Octave—Without MATLAB
Introducing pkg-fxp: a free, open-source fi-compatible fixed-point class
A Lesson in Statistics Using Random Sequences
Statistics may come naturally to some people, but it is a difficult subject for many of us. Using simulations helps to remove some of the mystery of statistics. Luckily, it is trivial to produce a pseudo-random Gaussian or uniform sequence: a single command in Matlab (or Python) does it. In this article, I try to explain a few concepts using Gaussian and uniform random sequences generated in Matlab. First, we’ll generate a Gaussian sequence, calculate its variance, and plot a histogram and probability density function (PDF). Next, we’ll simulate the roll of a single die to illustrate a uniform distribution. Then we’ll simulate rolling two dice and finally, as an illustration of the Central Limit Theorem, we’ll sum the total when rolling several dice to obtain an approximately Gaussian distribution.
Simple but Effective Spectrum Averaging
In this article, I provide a Matlab function that performs exponential PSD averaging, using first-order infinite impulse response (IIR) filtering to continuously average the PSD bins. This approach works well for computing the spectrum of a long-duration signal over time, because the spectrum is constantly updated as new PSD’s are computed. Conveniently, the time constant of the PSD averaging is determined by the single adjustable parameter α. I also provide a Matlab function for conventional (unweighted) PSD averaging. Neither function requires any canned code other than the Fast Fourier Transform (FFT), although I do use the Matlab hann window function for convenience.
The First-Order IIR Filter -- More than Meets the Eye
While we might be inclined to disdain the simple first-order infinite impulse response (IIR) filter, it is not so simple that we can’t learn something from it. Studying it can teach DSP math skills, and it is a very useful filter in its own right. In this article, we’ll examine the time response of the filter, compare the first-order IIR filter to the FIR moving average filter, use it to smooth a noisy signal, compute the functional form of the impulse response, and find the frequency response.
RF in Slow Motion: Sonifying a Wi-Fi7 Packet
What would a 160 MHz OFDM waveform up in the 5 GHz U-NII band sound like if scaled to audio frequencies to keep the same wavelength (acoustic vs RF)?
Off Topic: The True Gravitational Geodesic
The third of my off topic Physics series resulting in the true gravitational geodesic equation and some surprising results about gravity.
The Discrete Fourier Transform of Symmetric Sequences
Symmetric sequences arise often in digital signal processing. Examples include symmetric pulses, window functions, and the coefficients of most finite-impulse response (FIR) filters, not to mention the cosine function. Examining symmetric sequences can give us some insights into the Discrete Fourier Transform (DFT). An even-symmetric sequence is centered at n = 0 and xeven(n) = xeven(-n). The DFT of xeven(n) is real. Most often, signals we encounter start at n = 0, so they are not strictly speaking even-symmetric. We’ll look at the relationship between the DFT’s of such sequences and those of true even-symmetric sequences.
How the Cooley-Tukey FFT Algorithm Works | Part 4 - Twiddle Factors
The beauty of the FFT algorithm is that it does the same thing over and over again. It treats every stage of the calculation in exactly the same way. However, this. “one-size-fits-all” approach, although elegant and simple, causes a problem. It misaligns samples and introduces phase distortions during each stage of the algorithm. To overcome this, we need Twiddle Factors, little phase correction factors that push things back into their correct positions before continuing onto the next stage.
How the Cooley-Tukey FFT Algorithm Works | Part 3 - The Inner Butterfly
At the heart of the Cooley-Tukey FFT algorithm lies a butterfly, a simple yet powerful image that captures the recursive nature of how the FFT works. In this article we discover the butterfly’s role in transforming complex signals into their frequency components with efficiency and elegance. Starting with the 2-point DFT, we reveal how the FFT reuses repeated calculations to save time and resources. Using a divide-and-conquer approach, the algorithm breaks signals into smaller groups, processes them through interleaving butterfly diagrams, and reassembles the results step by step.
A Beginner's Guide To Cascaded Integrator-Comb (CIC) Filters
This article discusses the behavior, mathematics, and implementation of cascaded integrator-comb filters.
Free DSP Books on the Internet
While surfing the "net" I have occasionally encountered signal processing books whose chapters could be downloaded to my computer. I started keeping a list of those books and, over the years, that list has grown to over forty books. Perhaps the...
Handy Online Simulation Tool Models Aliasing With Lowpass and Bandpass Sampling
Analog Devices Inc. has posted a neat software simulation tool on their corporate web site that graphically shows the aliasing effects of both lowpass and bandpass periodic sampling. This is a nice tutorial tool for beginners in DSP. The...
A Differentiator With a Difference
Some time ago I was studying various digital differentiating networks, i.e., networks that approximate the process of taking the derivative of a discrete time-domain sequence. By "studying" I mean that I was experimenting with various...
Fixed-Point Simulation in GNU Octave—Without MATLAB
Introducing pkg-fxp: a free, open-source fi-compatible fixed-point class
A Fixed-Point Introduction by Example
Introduction The finite-word representation of fractional numbers is known as fixed-point. Fixed-point is an interpretation of a 2's compliment number usually signed but not limited to sign representation. It...
Minimum Shift Keying (MSK) - A Tutorial
Minimum Shift Keying (MSK) is one of the most spectrally efficient modulation schemes available. Due to its constant envelope, it is resilient to non-linear distortion and was therefore chosen as the modulation technique for the GSM cell phone...
Sinusoidal Frequency Estimation Based on Time-Domain Samples
The topic of estimating a noise-free real or complex sinusoid's frequency, based on fast Fourier transform (FFT) samples, has been presented in recent blogs here on dsprelated.com. For completeness, it's worth knowing that simple frequency estimation algorithms exist that do not require FFTs to be performed . Below I present three frequency estimation algorithms that use time-domain samples, and illustrate a very important principle regarding so called "exact" mathematically-derived DSP algorithms.
Understanding and Implementing the Sliding DFT
Introduction In many applications the detection or processing of signals in the frequency domain offers an advantage over performing the same task in the time-domain. Sometimes the advantage is just a simpler or more conceptually...
