## Second Order Discrete-Time System Demonstration

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 easily compute the time response to any input.  But note that the goal here is as much to...

## A Simplified Matlab Function for Power Spectral Density

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 all and how they default.

In this post, I create a simplified PSD function by putting a...

## Fractional Delay FIR Filters

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 number of taps, the group delay is always an integer + 0.5 samples.  Can we design a filter...

## Model Signal Impairments at Complex Baseband

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 complex-baseband signal.

I used a very simple complex-baseband model of a QAM system in my last

## Compute Modulation Error Ratio (MER) for QAM

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 diagram of a QAM system.  The transmitter includes a source of QAM symbols, a root-Nyquist...

## Plotting Discrete-Time Signals

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, which does not look like a sine when you connect the dots.  We typically want the plot of a...

## Interpolation Basics

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 values are close to the values of the continuous signal at the sample times [1].  For example, if...

## A Direct Digital Synthesizer with Arbitrary Modulus

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 entries long.   We’ll also demonstrate a Quadrature-output DDS.  A note on...

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 narrowband filters.

A biquad section block diagram using the Direct Form II structure [3,4] is...

## Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT

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:

$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$

Where

X(k) = discrete frequency spectrum of time sequence x(n)

## Canonic Signed Digit (CSD) Representation of Integers

February 18, 2017

In my last post I presented Matlab code to synthesize multiplierless FIR filters using Canonic Signed Digit (CSD) coefficients.  I included a function dec2csd1.m (repeated here in Appendix A) to convert decimal integers to binary CSD values.  Here I want to use that function to illustrate a few properties of CSD numbers.

In a binary signed-digit number system, we allow each binary digit to have one of the three values {0, 1, -1}.  Thus, for example, the binary value 1 1...

## Demonstrating the Periodic Spectrum of a Sampled Signal Using the DFT

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:

$$X(k)=\sum_{n=0}^{N-1}x(n)e^{-j2\pi kn/N} \qquad (1)$$

Where

X(k) = discrete frequency spectrum of time sequence x(n)

## Compute the Frequency Response of a Multistage Decimator

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 taps is lower.

The frequency response of the single-stage decimator before downsampling is just...

## Compute Modulation Error Ratio (MER) for QAM

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 diagram of a QAM system.  The transmitter includes a source of QAM symbols, a root-Nyquist...

## Design IIR Band-Reject Filters

In this post, I show how to design IIR Butterworth band-reject filters, and provide two Matlab functions for band-reject filter synthesis.  Earlier posts covered IIR Butterworth lowpass [1] and bandpass [2] filters.  Here, the function br_synth1.m designs band-reject filters based on null frequency and upper -3 dB frequency, while br_synth2.m designs them based on lower and upper -3 dB frequencies.   I’ll discuss the differences between the two approaches later in this...

## ADC Clock Jitter Model, Part 1 – Deterministic Jitter

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 transfers the jitter from the sample clock to the input signal.

## ADC Clock Jitter Model, Part 2 – Random Jitter

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 all the same phenomenon.  Typically, we call it jitter when we have a time-domain representation,...

## Matlab Code to Synthesize Multiplierless FIR Filters

This article presents Matlab code to synthesize multiplierless Finite Impulse Response (FIR) lowpass filters.

A filter coefficient can be represented as a sum of powers of 2.  For example, if a coefficient = decimal 5 multiplies input x, the output is $y= 2^2*x + 2^0*x$.  The factor of $2^2$ is then implemented with a shift of 2 bits.  This method is not efficient for coefficients having a lot of 1’s, e.g. decimal 31 = 11111.  To reduce the number of non-zero...