In this article, I try to explain a few concepts using Gaussian and uniform random...]]> Sat, 14 Mar 2026 17:02:00 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1773.php Digitally modulated signals and Gaussian noise are random signals, and randomness is apparent in their spectra.  For example, Figure 1 (left) shows the power spectral density (PSD) of a QAM-16 signal with added Gaussian noise.  While this PSD is perfectly correct, it has a large variation around the average values of the signal and the noise, making it difficult to estimate...]]> Fri, 02 Jan 2026 20:00:15 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1769.php 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,...]]> Sun, 16 Nov 2025 16:20:08 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1748.php  Half-band filters have -6 dB frequency of 1/4 the sample rate, and odd symmetry of the frequency response about 1/4 the sample rate.  Given the odd-symmetry of the response, the passband and stopband edge frequencies are symmetric with respect to fs/4.  This symmetry makes the halfband filter ideal for decimation by 2 or interpolation by 2.  And, remarkably, the...]]> Sun, 06 Jul 2025 18:35:32 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1718.php 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)...]]> Sun, 08 Dec 2024 12:57:57 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1696.php Discrete-time sequences arise in many ways:  a sequence could be a signal captured by an analog-to-digital converter; a series of measurements; a signal generated by a digital modulator; or simply the coefficients of a digital filter.  We may wish to know the frequency spectrum of any of these sequences.  The most-used tool to accomplish this is the Discrete Fourier...]]> Sat, 28 Sep 2024 17:36:07 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1642.php Sigma-delta digital-to-analog converters (SD DAC’s) are often used for discrete-time signals with sample rate much higher than their bandwidth.  For the simplest case, the DAC output is a single bit, so the only interface hardware required is a standard digital output buffer.  Because of the high sample rate relative to signal bandwidth, a very simple DAC reconstruction filter...]]> Sat, 16 Mar 2024 18:25:43 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1627.php As the name indicates, a digital to analog converter (DAC) converts a digital quantity to an analog voltage or current.  But more than this, a DAC converts a discrete-time signal into a continuous-time signal.  The latter operation is almost always accomplished by a zero-order hold (ZOH) function. As we’ll see, it is the ZOH that causes the sinx/x roll-off in the frequency...]]> Sun, 21 Jan 2024 18:29:17 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1609.php In my last post, I provided coefficients for several multiplierless half-band FIR filters [1].  In the comment section, Rick Lyons mentioned that such filters would be useful in a multi-stage decimator, as shown in Figure 1 for the decimate-by-8 case.  For such an arrangement, any subsequent multipliers save on resources, since they operate at 1/8th of the maximum sample frequency or...]]> Sun, 19 Nov 2023 18:42:45 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1585.php This article provides coefficients of multiplierless Finite Impulse Response 7-tap, 11-tap, and 15-tap half-band filters and Hilbert Transformers.  Since Hilbert transformer coefficients are simply related to half-band coefficients, multiplierless Hilbert transformers are easily derived from multiplierless half-bands.  Image attenuation of the Hilbert transformers presented here is...]]> Sat, 07 Oct 2023 14:14:47 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1542.php Designing an interpolator needn’t be confusing.  In this article, I’ll present a simple approach for designing interpolators that takes the guesswork out of determining the stopbands (For a basic introduction to interpolators, see my earlier post [1]).

Figure 1a shows a block diagram of an interpolator, which consists of an up-sampler that increases the sample rate of the input...]]> Thu, 06 Jul 2023 14:48:59 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1507.php There are many software applications that allow modeling LC filters in the frequency domain.  But sometimes it is useful to have a time domain model, such as when you need to analyze a mixed analog and DSP system.  For example, the system in Figure 1 includes an LC filter as well as a DSP portion.  The LC filter could be an anti-alias filter, a channel filter, or some other LC...]]> Wed, 19 Apr 2023 12:22:11 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1498.php The discrete frequency response H(k) of a Finite Impulse Response (FIR) filter is the Discrete Fourier Transform (DFT) of its impulse response h(n) [1].  So, if we can find H(k) by whatever method, it should be identical to the DFT of h(n).  In this article, we’ll find H(k) by using complex exponentials, and we’ll see that it is indeed identical to the DFT of h(n)....]]> Sat, 04 Feb 2023 19:01:05 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1487.php In this part, I’ll show how to design a Hilbert Transformer using the coefficients of a half-band filter as a starting point, which turns out to be remarkably simple.  I’ll also show how a half-band filter can be synthesized using the Matlab function firpm, which employs the Parks-McClellan algorithm.

A half-band filter is a type of lowpass, even-symmetric FIR filter having an odd...]]> Sun, 04 Dec 2022 19:19:19 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1486.php In some previous articles, I made use of the Hilbert transformer, but did not explain its theory in any detail.  In this article, I’ll dig a little deeper into how the Hilbert Transformer works.  Understanding the Hilbert Transformer involves a modest amount of mathematics, but the payoff in useful applications is worth it.

As we’ll learn, a Hilbert Transformer is just a...]]> Tue, 22 Nov 2022 14:41:43 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1454.php What is Mathematics is a classic, lucidly written survey of mathematics by Courant and Robbins.  The first edition was published in 1941!  I have only read a portion of it, mainly the chapter on calculus.  One page of Courant is worth about five pages of my old college calculus textbook, and it’s a lot more fun to read.

The reader of this book should already be familiar with...]]> Mon, 20 Jun 2022 14:40:09 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1447.php When it comes to noise, all differentiators are not created equal.  Figure 1 shows the magnitude response of two differentiators.  They both have a useful bandwidth of a little less than π/8 radians (based on maximum magnitude response error of 2%).  Suppose we apply a signal with Gaussian noise to each of these differentiators.  The sinusoidal signal with noise is shown...]]> Mon, 28 Mar 2022 19:01:01 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1441.php We don’t often think about signal transmission lines, but we use them every day.  Familiar examples are coaxial cable, Ethernet cable, and Universal Serial Bus (USB).  Like it or not, high-speed clock and signal traces on printed-circuit boards are also transmission lines.

While modeling transmission lines is in general a complex undertaking, it is surprisingly simple to model a...]]> Wed, 12 Jan 2022 19:58:39 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1433.php The Discrete Fourier Transform (DFT) is used to find the frequency spectrum of a discrete-time signal.  A computationally efficient version called the Fast Fourier Transform (FFT) is normally used to calculate the DFT.  But, as many have found to their dismay, the FFT, when used alone, usually does not provide an accurate spectrum.  The reason is a phenomenon called spectral...]]> Mon, 15 Nov 2021 13:18:30 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1418.php Digitizing a signal using an Analog to Digital Converter (ADC) usually requires an anti-alias filter, as shown in Figure 1a.  In this post, we’ll develop models of lowpass Butterworth and Chebyshev anti-alias filters, and compute the time domain and frequency domain output of the ADC for an example input signal.  We’ll also model aliasing of Gaussian noise.  I hope the...]]> Sun, 26 Sep 2021 12:11:57 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1404.php

Lyons, Richard G., Understanding Digital Signal Processing, Third Ed., Prentice Hall, 2011, section 13.7.

            Neil Robertson      June,...]]> Fri, 18 Jun 2021 17:58:51 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1398.php When simulating digital receivers, we often want to check performance with added Gaussian noise.  In this article, I’ll derive the simple equations for the rms noise level needed to produce a desired carrier to noise ratio (CNR or C/N).  I also provide a short Matlab function to generate a noise vector of the desired level for a given signal vector.

The...]]> Sun, 11 Apr 2021 15:12:57 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1387.php Perhaps we should call most Power Spectral Density (PSD) calculations relative PSD, because usually we don’t have to worry about absolute power levels.  However, for cases (e.g., measurements or simulations) where we are concerned with absolute power, it would be nice to be able to display it on a PSD plot.  Unfortunately, you can’t read the power directly from the plot. ...]]> Sun, 07 Feb 2021 15:25:26 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1380.php When a sinewave is applied to a data converter (ADC or DAC), device nonlinearities produce harmonics.  If a harmonic frequency is greater than the Nyquist frequency, the harmonic appears as an alias.  In this case, it is not at once obvious if a given spur is a harmonic, and if so, its order.  In this article, we’ll present Matlab code to simulate the data converter...]]> Mon, 11 Jan 2021 18:12:36 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1375.php Cascade-Integrator-Comb (CIC) filters are efficient fixed-point interpolators or decimators.  For these filters, all coefficients are equal to 1, and there are no multipliers.  They are typically used when a large change in sample rate is needed.  This article provides two very simple Matlab functions that can be used to compute the spectral images of CIC interpolators and...]]> Sun, 01 Nov 2020 17:50:34 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1361.php In his book on multirate signal processing, harris presents a nifty technique for designing square-root Nyquist FIR filters with good stopband attenuation [1].  In this post, I describe the method and provide a Matlab function for designing the filters.  You can find a Matlab function by harris for designing the filters at [2].

Single-carrier modulation, such as QAM,...]]> Mon, 13 Jul 2020 18:20:47 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1355.php

In the following analysis, we assume that signal phase at the amplifier output is not a function of amplitude.  With this assumption, the output y of a non-ideal...]]> Tue, 09 Jun 2020 16:27:59 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1341.php Discrete-time systems are remarkable:  the time response can be computed from mere difference equations, and the coefficients ai, bi of these equations are also the coefficients of H(z).  Here, I try to illustrate this remarkableness by converting a continuous-time second-order system to an approximately equivalent discrete-time system.  With a discrete-time model, we can then...]]> Wed, 01 Apr 2020 18:20:59 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1333.php In an earlier post [1], I showed how to compute power spectral density (PSD) of a discrete-time signal using the Matlab function pwelch [2].  Pwelch is a useful function because it gives the correct output, and it has the option to average multiple Discrete Fourier Transforms (DFTs).  However, a typical function call has five arguments, and it can be hard to remember how to set them...]]> Tue, 03 Mar 2020 18:06:17 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1327.php Consider the following Finite Impulse Response (FIR) coefficients:

b = [b0 b1 b2 b1 b0]

These coefficients form a 5-tap symmetrical FIR filter having constant group delay [1,2] over 0 to fs/2 of:

D = (ntaps – 1)/2 = 2      samples

For a symmetrical filter with an odd number of taps, the group delay is always an integer number of samples, while for one with an even...]]> Sun, 09 Feb 2020 20:15:03 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1312.php In this article, we develop complex-baseband models for several signal impairments: interfering carrier, multipath, phase noise, and Gaussian noise.  To provide concrete examples, we’ll apply the impairments to a QAM system. The impairment models are Matlab functions that each use at most seven lines of code.  Although our example system is QAM, the models can be used for any...]]> Wed, 11 Dec 2019 15:45:53 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1305.php This post defines the Modulation Error Ratio (MER) for QAM signals, and shows how to compute it.  As we’ll see, in the absence of impairments other than noise, the MER tracks the signal’s Carrier-to-Noise Ratio (over a limited range).  A Matlab script at the end of the PDF version of this post computes MER for a simplified QAM-64 system.

Figure 1 is a simplified block...]]> Tue, 05 Nov 2019 18:54:37 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1298.php A discrete-time sinusoid can have frequency up to just shy of half the sample frequency.  But if you try to plot the sinusoid, the result is not always recognizable.  For example, if you plot a 9 Hz sinusoid sampled at 100 Hz, you get the result shown in the top of Figure 1, which looks like a sine.  But if you plot a 35 Hz sinusoid sampled at 100 Hz, you get the bottom graph,...]]> Sun, 15 Sep 2019 15:12:03 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1293.php This article covers interpolation basics, and provides a numerical example of interpolation of a time signal.  Figure 1 illustrates what we mean by interpolation.  The top plot shows a continuous time signal, and the middle plot shows a sampled version with sample time Ts.  The goal of interpolation is to increase the sample rate such that the new (interpolated) sample...]]> Tue, 20 Aug 2019 20:14:46 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1275.php Suppose you have a system with a 10 MHz sample clock, and you want to generate a sampled sinewave at any frequency below 5 MHz on 500 kHz spacing; i.e., 0.5, 1.0, 1.5, … MHz.  In other words, f = k*fs/20, where k is an integer and fs is sample frequency.  This article shows how to do this using a simple Direct Digital Synthesizer (DDS) with a look-up table that is at most 20...]]> Mon, 03 Jun 2019 20:05:10 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1257.php In an earlier post [1], we implemented lowpass IIR filters using a cascade of second-order IIR filters, or biquads.  

This post provides a Matlab function to do the same for Butterworth bandpass IIR filters.  Compared to conventional implementations, bandpass filters based on biquads are less sensitive to coefficient quantization [2].  This becomes important when designing...]]> Sat, 20 Apr 2019 19:18:19 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1235.php One of the basic DSP principles states that a sampled time signal has a periodic spectrum with period equal to the sample rate.  The derivation of can be found in textbooks [1,2].  You can also demonstrate this principle numerically using the Discrete Fourier Transform (DFT).

The DFT of the sampled signal x(n) is defined as:

$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N}...]]> Sat, 09 Mar 2019 19:40:12 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1228.php Figure 1a shows the block diagram of a decimation-by-8 filter, consisting of a low-pass finite impulse response (FIR) filter followed by downsampling by 8 [1].  A more efficient version is shown in Figure 1b, which uses three cascaded decimate-by-two filters.  This implementation has the advantages that only FIR 1 is sampled at the highest sample rate, and the total number of filter...]]> Sun, 10 Feb 2019 20:32:27 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1221.php In my last post, we saw that finding the spectrum of a signal requires several steps beyond computing the discrete Fourier transform (DFT)[1].  These include windowing the signal, taking the magnitude-squared of the DFT, and computing the vector of frequencies.  The Matlab function pwelch [2] performs all these steps, and it also has the option to use DFT averaging to compute the...]]> Sun, 13 Jan 2019 18:18:48 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1211.php The Discrete Fourier Transform (DFT) operates on a finite length time sequence to compute its spectrum.  For a continuous signal like a sinewave, you need to capture a segment of the signal in order to perform the DFT.  Usually, you also need to apply a window function to the captured signal before taking the DFT [1 - 3].  There are many different window functions and each...]]> Tue, 18 Dec 2018 20:43:38 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1191.php This post provides a Matlab function that designs linear-phase FIR sinx/x correctors.  It includes a table of fixed-point sinx/x corrector coefficients for different DAC frequency ranges.

A sinx/x corrector is a digital (or analog) filter used to compensate for the sinx/x roll-off inherent in the digital to analog conversion process.  In DSP math, we treat the digital signal...]]> Sun, 22 Jul 2018 17:22:49 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1177.php Sometimes you may need to phase-lock a numerically controlled oscillator (NCO) to an external clock that is not related to the system clocks of your ASIC or FPGA.  This situation is shown in Figure 1.  Assuming your system has an analog-to-digital converter (ADC) available, you can sync to the external clock using the scheme shown in Figure 2.  This time-domain PLL model is...]]> Sun, 27 May 2018 18:06:07 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1160.php In Part 1, I presented a Matlab function to model an ADC with jitter on the sample clock, and applied it to examples with deterministic jitter.  Now we’ll investigate an ADC with random clock jitter, by using a filtered or unfiltered Gaussian sequence as the jitter source.  What we are calling jitter can also be called time jitter, phase jitter, or phase noise.  It’s...]]> Sun, 22 Apr 2018 17:30:15 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1157.php Analog to digital converters (ADC’s) have several imperfections that affect communications signals, including thermal noise, differential nonlinearity, and sample clock jitter [1, 2].  As shown in Figure 1, the ADC has a sample/hold function that is clocked by a sample clock.  Jitter on the sample clock causes the sampling instants to vary from the ideal sample time.  This...]]> Mon, 16 Apr 2018 17:29:19 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1147.php In this article, we’ll describe how to use a Hilbert transformer to make a phase shifter or frequency shifter.  In either case, the input is a real signal and the output is a real signal.  We’ll use some simple Matlab code to simulate these systems.  After that, we’ll go into a little more detail on Hilbert transformer theory and design. 

A...]]> Sun, 25 Mar 2018 16:43:41 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1141.php In this article, we’ll show how to compute the coefficients that result when you cascade discrete-time systems.  With the coefficients in hand, it’s then easy to compute the time or frequency response.  The computation presented here can also be used to find coefficients of mixed discrete-time and continuous-time systems, by using a discrete time model of the continuous-time...]]> Sun, 04 Mar 2018 16:08:17 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1137.php This article shows how to implement a Butterworth IIR lowpass filter as a cascade of second-order IIR filters, or biquads.  We’ll derive how to calculate the coefficients of the biquads and do some examples using a Matlab function biquad_synth provided in the Appendix.  Although we’ll be designing Butterworth filters, the approach applies to any all-pole lowpass filter...]]> Sun, 11 Feb 2018 21:08:34 +0000 Neil Robertson https://www.dsprelated.com/showarticle/1135.php This post is the fourth in a series of tutorials on IIR Butterworth filter design.  So far we covered lowpass [1], bandpass [2], and band-reject [3] filters; now we’ll design highpass filters.  The general approach, as before, has six steps: