DSP Archives | Page 6 of 14

Odd symmetry around frequency points at half symbol rate adding up to a flat spectrum

February 29, 2024 DSP / Wireless/SDR

Proof of Poisson Sum Formula

The Poisson sum formula was discovered by the French mathematician and physicist Siméon Denis Poisson. It has several applications in digital signal processing, among which our concern is the periodic summation of modulated pulses in digital communication systems. Assume that $p(t)$ is a pulse shape (or any continuous-time function if you are not familiar with digital communications) and $P(f)$ is its Fourier Transform. The pulse is sampled at a rate of $f_s$ to produce its discrete version $p(nT_s)$ where $T_s=1/f_s$ is the duration between two samples. The Poisson summation formula relates these two quantities as \begin{equation}\label{equation-poisson-sum-formula} \frac{1}{T_s}\sum _{k=-\infty}^{\infty} P\left(f+\frac{k}{T_s}\right) =

July 19, 2023 DSP / Wireless/SDR

Antenna Arrays: Whole is More than its Parts

In a previous article, we described how multiples antennas can be viewed as sampling the signal in space domain in a direct analogy to sampling the signal in time domain. Filters in discrete-time domain are usually defined by numerical values in the time axis. Space allows three-dimensional freedom. A filter in discrete space can have its elements placed anywhere in the space. This sounds very complicated! But we will show it is not. A Simple Time Domain Filter: 2-Tap FIR The simplest discrete FIR filter has only two samples of unit magnitude. Assuming unit magnitude and zero phase for both

A bank of N filters each centered at discrete frequency k/N

March 19, 2020 DSP

Discrete Fourier Transform (DFT) as a Filter Bank

We have discussed before what a Discrete Fourier Transform (DFT) is and how to find the DFT of some commonly used signals. Here, we will see how a DFT acts as a (crude) bank of filters that can pass the signal contents around a desired frequency while blocking the rest. Let us start with the definition of the DFT. \begin{equation*} \begin{aligned} S_I[k]\: &= \sum \limits _{n=0} ^{N-1}\left[ s_I[n] \cos 2\pi\frac{k}{N}n + s_Q[n] \sin 2\pi\frac{k}{N}n\right] \\ S_Q[k] &= \sum \limits _{n=0} ^{N-1}\left[ s_Q[n] \cos 2\pi\frac{ k}{N}n – s_I[n] \sin 2\pi\frac{k}{N}n\right] \end{aligned} \end{equation*} for each $k$. In complex notation, this DFT is

June 16, 2023 DSP

3 Ways to Approximate atan2( ) in Hardware

When we think about signal processing, the focus is usually magnitude response of a system. However, in several DSP applications, the signal phase holds as much, if not more, significance as the magnitude response. For example, in digital FM demodulation, carrier phase synchronization and RF ranging, the phase (found through arctangent in four quadrants) of a complex signal needs to be computed by an FPGA or a DSP for further processing. In image processing applications, such an operation is also required to calculate the gradient orientations used in several popular feature descriptors like the Scale-Independent Feature Transform (SIFT) or the

Farrow structure for cubic interpolation

January 11, 2022 DSP

Fractional Delay Filters Using the Farrow Structure

In the discussion on piecewise polynomial interpolation, we emphasized on the fact that the fractional interval $\mu_m$ needs to be updated for each symbol time $mT_M$ and hence the subscript $m$ in $\mu_m$. For this reason, the interpolation process becomes a two-step procedure. Update the filter coefficients $h_p[n]$. Perform the convolution between $z(nT_S)$ and $h_p[n]$. This process can be simplified if the two steps above can be combined in such a way that $\mu_m$ update is weaved into the convolution operation. In other words, instead of a two-input hardware multiplication with two variable quantities, complexity can be reduced by restructuring