DSPIllustrations.com

The Prime Music, Staircase and Riemann Hypothesis

Maximilian Matthe — Sun, 02 Sep 2018 01:00:00 +0200

The Prime Numbers' Music, Staircase and Riemann Hypothesis¶

For DSP, mathematics is a fundamental tool to understand the different techniques. Hence, I'm also interested in different aspects of mathematics. And what aspect could be more "pure" mathematics than number theory and prime numbers? In this regard, to at least understand the still unsolved Riemann Hypothesis, where a price money of 1,000,000 will be given to the one who proves or disproves it, has been bothering me for quite some time. Despite a lot of material about the hypothesis being online, it was the book of Barry Mazur and William Stein entitled "Prime Numbers and the Riemann Hypothesis" which provided me at least a superficial understanding of it.

The nice thing about this book is its very simple approach, requiring not more than high school math. Each page contains graphs and illustrations of the analyzed functions and hence makes it quite easy to follow. Accordingly, it's a very nice book to get to know a bit about the Riemann Hypothesis (RH). However, admittedly, the mathematical treatment is intentionally quite superficial.

In this notebook, I do not want to directly talk about the RH. Instead, the notebook arose when I tried to replicate some calculations from the ...

Schmidl&Cox Synchronization for OFDM

Maximilian Matthe — Thu, 10 May 2018 23:55:00 +0200

The Schmidl & Cox Synchronization Technique for OFDM¶

tl;dr: Given some received OFDM signal like the following, how can one know at which point in time the OFDM symbols are located? Or, equivalently, on which signal part the receiver needs to perform the FFT?

In the OFDM example, we have described the OFDM modulation and demodulation, including channel estimation and CP insertion. On the significance of the CP we have already elaborated in another article. However, in all these works, we have assumed that the receiver knows, at which point in time the OFDM symbol is received and hence on which received samples to perform the FFT.

However, in reality this information is not available by default. Instead, the receiver needs to perform a synchronization procedure to obtain the start of the OFDM symbol. When talking about OFDM, the most fundamental work was published by Timothy Schmidl and Donald Cox in their paper Robust Frequency and Timing Synchronization for OFDM. In this notebook, we are going to illustrate their algorithm presented in this paper concerning the ...

Using your soundcard for hands-on digital communication

Maximilian Matthe — Fri, 09 Feb 2018 23:55:00 +0100

Using your Soundcard as a Software-defined Radio¶

Getting hands-on experience on wireless communications¶

Do you always wonder how you can apply the things you learned in lectures in real-world experiments?

Are you bored of just simulating things without real-world applications?

Have you ever wondered, how wireless communications really work, despite all the simulations and math?

Use a cheap soundcard as a low-cost software-defined radio!

With a soundcard, a speaker and a microphone you have all the things necessary for a wireless transmission system:

The speaker mimics the transmit antenna.
The soundcard performs analog/digital conversion and sampling.
The microphone equals the receive antenna.

Using your soundcard to study wireless communications has several advantages:

Soundcards are cheap, you can get a simple USB sound card for below 10 bucks.
With an audio sampling rate of 44.1kHz, the amount of data is small enough to be processed on a normal PC without the need for powerful FPGA or other resources.
As the transmit signal is actual sound, you ...

Intuitive Explanation of OFDM

Maximilian Matthe — Sun, 23 Apr 2017 14:00:00 +0200

An intuitive explanation on how OFDM works¶

This article is about the fundamental principle of OFDM. Without going into the mathematical details, I'm going to explain, why the commonly used OFDM waveform is designed as it is. In a previous article, we have already shown a source code example of OFDM.

Let us start with the assumption that we face a static multipath channel (static means, the channel coefficients do not change over all times, i.e. it is time-invariant). Such a channel is a linear, time-invariant (LTI) system, and hence it is described solely by its impulse response h(t). The output signal y(t) of this system is the convolution of the input signal x(t) with the impulse response h(t):

Eigenfunctions of the time-invariant channel¶

The most important property of LTI systems is, that complex exponentials of any frequency are eigenfunctions of the system. Literally, if the input to an LTI system is a complex exponential, its output will be a complex exponential of the same frequency, but with a different amplitude. Let's verify ...

Linearity, Causality and Time-Invariance of a System

Maximilian Matthe — Thu, 13 Apr 2017 22:35:00 +0200

Linearity, Causality and Time-Invariance of a System¶

The notion of a system is central in digital communications and particularly system's theory. Abstractly, a system is defined as something that takes an input signal and produces an output signal by some transformation rule Tr. y(t)=Tr\{x(t)\}

Many relations in the real world can actually be understood as a system. Some examples include:

You press a key on your keyboard, and the corresponding letter appears on your screen. What happens if you press two letters at the same time? Is this system "linear"?
You speak into your microphone, and it converts your voice into electrical current. Hopefully this system does not introduce a lot of distortion.
You inflate the tire of your bike. It responds with the pressure in the tire. The pressure can be seen as the summation of all the air that has flown into and out of the tire.

Let's take a more abstract example: A system can amplify the input signal, by doubling its amplitude: y(t)=2x(t) Let us illustrate this very simple system with some code. As an example, we take the input signal x(t)=\sin(t). Then, we can straight-forwardly implement the system as taking a function (a signal) ...

The complex Fourier Series and its relation to the Fourier Transform

Maximilian Matthe — Tue, 11 Apr 2017 22:00:00 +0200

The complex Fourier Series and its relation to the Fourier Transform¶

In two recent articles we have talked about the Fourier Series and an application in harmonic analysis of instrument sounds in terms of their Fourier coefficients. In this article, we will analyze the relation between the Fourier Series and the Fourier Transform.

The Fourier Series as sums of sines and cosines¶

To recap, the Fourier series of a signal x(t) with period P is given by

\begin{align}x(t)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(2\pi nt/P)+b_n\sin(2\pi nt/P)\end{align}

where the coefficients are given by

\begin{align}a_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\cos(2\pi nt/P)dt\\b_n&=\frac{2}{P}\int_{-\frac{P}{2}}^\frac{P}{2}x(t)\sin(2\pi nt/P)dt\end{align}.

As we see, the Fourier series is a sum of sines and cosines with different amplitudes. Let us first look at the sum of a sine and cosine with different amplitudes:

Sum of a sine and cosine with equal frequency¶

{% includeTrinket "a9b2abf8de" "FourierSeriesAndFourierTransform_0" %}

Fs = 100  # the sampling frequency for the discrete analysis
T = 3     # time duration to look at
P = 1     # signal period
t = ...

The Sound of Harmonics - Approximating instrument sounds with Fourier Series

Maximilian Matthe — Tue, 28 Mar 2017 23:50:00 +0200

The Sound of Harmonics - Approximating instruments with Fourier Series¶

In a previous article about the Fourier Series calculation we have illustrated how different numbers of harmonics approximate artificial periodic functions. The present post applies the results to the analysis of instrument sounds, namely sounds of a saxophone.

When playing a stationary tone on a saxophone, we hear a constant sound. Hence, we can assume its waveform is periodic, since we could start to listen to the tone at any time and would still hear the same tone. So, the waveform needs to repeat itself over and over again. In this case, it should be possible to expand the waveform into sines and cosines of harmonic frequencies and reconstruct the original signal from them.

We want to verify this is assumption with this post. Let us start with functions to calculate the Fourier series, fourierSeries and for reconstructing a signal from its Fourier series coefficients, reconstruct.

def fourierSeries(period, N):
    """Calculate the Fourier series coefficients up to the Nth harmonic"""
    result = [] ...

Fourier Series and Harmonic Approximation

Maximilian Matthe — Sat, 18 Mar 2017 23:55:00 +0100

The Fourier Series and Harmonic Approximation¶

In this article, we will walk through the origins of the Fourier transform: the Fourier Series. The Fourier series takes a periodic signal x(t) and describes it as a sum of sine and cosine waves. Noting that sine and cosine are themselves periodic functions, it becomes clear that x(t) is also a periodic function.

Mathematically, the Fourier series is described as follows. Let x(t) be a periodic function with period T, i.e.

x(t)=x(t+nT), n\in\mathbb{Z}.

Then, we can write x(t) as a Fourier series by

x(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(2\pi \frac{nt}{T})+b_n\sin(2\pi\frac{nt}{T}),

where a_n and b_n are the coefficients of the Fourier series. They can be calculated by \begin{align}a_n&=\frac{2}{T}\int_0^Tx(t)\cos(2\pi \frac{nt}{T})dt\\ b_n&=\frac{2}{T}\int_0^Tx(t)\sin(2\pi \frac{nt}{T})dt\end{align}.

Note that for a function with period T, the frequencies of the sines and cosines are \frac{1}{T}, \frac{2}{T}, \frac{3}{T}, \dots, i.e. they are multiples of the fundamental frequency \frac{1}{T}, which is the inverse period duration of the function. Therefore the frequency \frac{n}{T} is called the nth harmonic. The name harmonic stems from the fact for the human ear frequencies with integer ratios sound "nice", and the frequencies are all integer multiples of the fundamental frequency.

Let us verify the calculation of the Fourier coefficients and the function reconstruction numerically. First, we ...

The Convolution Theorem and Application Examples

Maximilian Matthe — Sun, 12 Mar 2017 00:00:00 +0100

The Convolution Theorem with Application Examples¶

The convolution theorem is a fundamental property of the Fourier transform. It is often stated like

"Convolution in time domain equals multiplication in frequency domain"

or vice versa

"Multiplication in time equals convolution in the frequency domain"

In this notebook we will illustrate what that means by pictorial examples. First, let us state the first version of the theorem mathematically. Let x(t) and y(t) be two arbitrary signals. Then, we have

\mathcal{F}\{x(t)*y(t)\}(f)=\mathcal{F}\{x(t)\}(f)\cdot\mathcal{F}\{y(t)\}(f),

where \mathcal{F}\{x(t)\}(f) denotes the Fourier transform of x(t), evaluated at the frequency f. In other words, we can say:

"The spectrum of the convolution two signals equals the multiplication of the spectra of both signals"

Let us first recap convolution (a more detailed description is given in another article on convolution): Given two signals x(t) and y(t), their convolution is defined by z(t)=x(t)*y(t)=\int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau.

Literally, we take one signal, mirror it in time and shift it in time domain. Then we multiply this signal with the other signal and calculate the integral of the overlapping part. Let us now calculate the convolution of two arbitrary signals and look at the result in time and frequency domain. Here, ...

Multipath Propagation and its Effect on Audio

Maximilian Matthe — Mon, 06 Mar 2017 21:24:00 +0100

Multipath Propagation and Multipath Channel¶

In this article, we will examine the effect of multipath propagation to a transmitted signal. Furthermore, we will illustrate the effect with the help of audio examples (scroll down for the examples).

First, let's try to understand the transmission of a signal from a source S to a destination D, which are d meters apart:

If we assume, the signal travels with velocity c, it will take \tau=\frac{d}{c} seconds until it arrives at the destination. Mathematically, we have

y(t)=x\left(t-\frac{d}{c}\right),

where x(t) is the transmitted signal and y(t) is the received signal. Now, we know that the signal from the source is not only transmitted into the direction of D, but in all directions. Let's assume somewhere around the source, there is a big wall that can reflect the signal:

So, at the destination we receive two copies of ...

The Cyclic Prefix (CP) in OFDM

Maximilian Matthe — Fri, 03 Mar 2017 04:13:00 +0100

The Cyclic Prefix for OFDM¶

In a previous post, we have elaborated about the building blocks of OFDM.

There, we have stated two benefits of using a cyclic prefix between subsequent OFDM symbols:

The CP isolates different OFDM blocks from each other when the wireless channel contains multiple paths, i.e. is frequency-selective.
The CP turns the linear convolution with the channel into a circular convolution. Only with a circular convolution, we can use the single-tap equalization OFDM is so famous for.

In the following, we will elaborate on these two aspects. First, let us define some parameters for our OFDM system:

N = 128     # OFDM block length (i.e. subcarrier count)
NCP = 32    # Length of OFDM CP

ofdm1 = np.arange(N)                      # generate some arbitrary signal for one OFDM block
ofdm2 = 2* ...

How does Quantization Noise Sound?

Maximilian Matthe — Mon, 27 Feb 2017 00:00:00 +0100

How does Quantization Noise sound?¶

In a last article, we explained the mathematical effect of quantization and what the resulting quantization noise is. In this article, we will hear, how the quantization noise actually sounds. As a teaser, listen to the following:

# For running this code, the code snippets below need to be run beforehand
display(HTML("Original signal:" + Audio(data=data_music, rate=rate)._repr_html_()))
showQuantization(data_music, U=1,bits=4, showSignals=False);

Original signal: Your browser does not support the audio element.

Quantized to q=4 bitsQuantization Noise ...

Circular Convolution Example

Maximilian Matthe — Wed, 22 Feb 2017 00:00:00 +0100

Circular Convolution¶

In a previous post, we have explained the importance of the convolution operation for signal processing and signal analysis. We have described the convolution integral and illustrated the involved functions.

In this post we will focus on an operation called Circular convolution which is strongly related to the conventional convolution (also called linear convolution) we have described before. Let us reconsider the normal, linear convolution in the discrete domain. Given two sequences x[n] and h[n], their convolution is given by

(x*h)[n] = \sum_{n'=-\infty}^{\infty}x[n']\cdot h[n-n'], \quad n=-\infty,\dots,\infty.

The linear convolution lets one one sequence slide over the other and sums the overlapping parts. The circular convolution of two sequences x[n], h[n] is now considering a wrap-around of the sequences after a period of N samples. So, the circular convolution is defined by

(x\otimes h)[n]=\sum_{n'=0}^{N-1}x[n']\cdot h[(n'-n)_N], \quad n=0,\dots,N-1,

where (n'-n)_N gives the remainder of n'-n divided by N. For example, (-1)_N=N-1. This means, if the index for x[(n'-n)_N] would leave the range 0,\dots,N-1 to the left, it would wrap around and come in from the right again. This means that the circular convolution is periodic with length N.

In discrete domain, the convolution theorem actually holds only for ...

Quantization and Quantization noise

Maximilian Matthe — Fri, 17 Feb 2017 00:00:00 +0100

Quantizers and Quantization Noise¶

A quantizer is a signal processing block, that maps a continuous amplitude to a discrete amplitude. The output of the quantizer is discrete, meaning that it can only output Q different values. Practically, the quantizer is an analog-to-digital converter, since it maps the continuous input amplitude to a digital representation of this value. Formally, the quantized output Q[x] of some input value x, is given by

Q[x] = \arg \min_{l\in\mathcal{S}} |l-x|.

Here, the set \mathcal{S} contains all possible output values of the quantizer, which we name quantization levels. What does that mean? It means, that for a given input x, the quantizer returns the quantization level l, which is closest to the input value. Hence, a quantizer is completely defined by its set of quantization levels \mathcal{S}.

Let's assume the quantizer should be able to quantize values between the input amplitudes -U and +U, i.e. the peak-to-peak amplitude range equals 2U. Furthermore, the quantization levels should be encoded by b bits. This gives us a number of 2^q different quantization levels in \mathcal{S}. Then, a logical decision is to make the distance \Delta_s between the quantization levels constant, and equal to

\Delta_s = \frac{2U}{q}.

Commonly, there are two different quantizers used, the ...

Baseband up- and downconversion and IQ modulation

Maximilian Matthe — Thu, 16 Feb 2017 00:00:00 +0100

Baseband signal upconversion and IQ Modulation and Demodulation¶

In this article, we will go through the basic steps of the up- and downconversion of a baseband signal to the passband signal. In most digital signal processing devices, any signal processing is performed in the baseband, i.e. where the signals are centered around the DC frequency. These baseband signals are mainly complex-valued. However, only real-valued signals can be sent with real-world physical devices. The process of upconversion thus has two purposes:

Convert the complex-valued baseband signal to a real-valued signal which can be transmitted over an antenna, cable or similar.
Adapt the transmit signal such that it uses a specific frequency band of the physical channel. This way, multiple signals can be transmitted independently on different frequency bands.

At the receiver, the received signal is then downconverted to baseband such that the subsequent processing can be done in complex-valued baseband domain.

A generic digital transceiver system with digital baseband processing, analog to digital conversion and up- and downconversion is presented in the figure below:

{% include "references/digitalmodulationscourse.html" %}

In the ...

Convolution Examples and the Convolution Integral

Maximilian Matthe — Wed, 15 Feb 2017 00:00:00 +0100

Convolution Examples and the Convolution Integral¶

In this notebook, we will illustrate the convolution operation. The convolution of two signals is a fundamental operation in signal processing. Mainly, because the output of any linear time-invariant (LTI) system is given by the convolution of its impulse response with the input signal. Another important application of convolution is the convolution theorem, which states that multiplication in time-domain corresponds to convolution in frequency domain and vice versa.

In this notebook, we will illustrate the operation of convolution and how we can calculate it numerically. Formally, the convolution (f_1*f_2)(t) of two signals f_1(t) and f_2(t) is defined by the convolution integral

(f_1*f_2)(t) = \int_{-\infty}^{\infty}f_1(\tau)f_2(t-\tau)d\tau.

So, the convolution of two function is the integral over the product of both functions, where one function is time-shifted and flipped in time. Let us not think about why this operation makes sense for now. Instead, let's define two functions f_1(t) and f_2(t):

f1 = lambda t: np.maximum(0, 1-abs(t))
f2 = lambda t: (t>0) ...

Aliasing and Anti-Aliasing Filter

Maximilian Matthe — Tue, 14 Feb 2017 00:00:00 +0100

Aliasing and Anti-Aliasing Filters¶

In this article we will explain the effect of aliasing, why it occurs and what can be done against it. Aliasing can occur whenever a sample rate conversion or sampling of an analog signal is performed. Great care needs to be taken to inhibit aliasing, as otherwise the resulting signal can be severely degraded.

To motivate this, let us listen to the effect of aliasing for three different kinds of signals. We take an original audio signal and perform a downsampling operation, for example to save bandwidth. Naturally, higher frequencies in the signal cannot be represented accurately with a too low sampling rate, so the downsampled audio does not sound as "bright" as the original. Despite, the effect of aliasing even impedes the audio quality. For each signal, first the original signal is presented, followed by a downsampled version that contains aliasing. Finally, the downsampled audio that has been sent through an anti-aliasing filter to prevent aliasing is provided.

A speech signal (taken from LibriVox). Here, aliasing manifests itself as some extra noise, especially around characters like 's' and 'z'.

# to call this function, the code below needs to be executed before
 ...

Group delay and phase delay example

Maximilian Matthe — Tue, 07 Feb 2017 00:00:00 +0100

Group delay and phase delay example¶

In this notebook, we are going to explore the meaning of the group delay \tau_g and the phase delay \tau_\phi of a filter. For this purpose, we will investigate the properties of a bandpass series RLC filter, due to its simplicity and wide applicability.

The group delay and phase delay of a filter are characterizations of its transfer function. Assume a filter with some arbitrary impulse response h(t). We know, when a signal x(t) is sent through the filter, the filter output y(t) is given by

y(t)=h(t)*x(t),

i.e. the impulse response is convolved with the input signal to yield the output signal. Now, to understand the group delay and phase delay, let us consider that the input signal x(t) consists of a carrier wave with frequency f_0 and an envelope function a(t):

x(t)=a(t)\sin(2\pi f_0t).

Given that a(t) changes much slower than its carrier, the output of the filter is approximately given by

y(t)=h(t)*x(t)\approx |H(f_0)| a(t-\tau_g)\sin(2\pi f_0(t-\tau_\phi)).

Here, \tau_g and \tau_\phi denote the group and phase delay of the filter. This means, that an envelope function that is modulated with a given carrier frequency is delayed by the group delay \tau_g and amplified by the frequency response of the filter at f_0 and the carrier ...

Decibel conversion: Factor 10 or Factor 20?

Maximilian Matthe — Wed, 01 Feb 2017 00:00:00 +0100

Decibel Conversion: Factor 10 or 20?¶

The decibel is used in a wide range of applications. Decibels are especially used, when a referring to power or a derived measure, which values can vary in a wide range. The most prominent usage of decibels is in sound volume. So, for example a sound of 0dB is barely hearable, whereas a vaccum cleaner on average has 75dB and a rock concert reaches about 110dB.

In class, you often find the following definitions for something in decibels

X_{dB}=10\log_{10}\left(\frac{X_{lin}}{X_{ref}}\right).

This equation transforms quantity X_{lin} from linear scale to a quantity in dB scale X_{dB}. In order to do that, first the linear quantity is related to a reference quantity X_{ref} and the ratio of both is transformed into the log-domain. Apparently, X_{dB} does actually have no unit and we artificially add dB to make clear we are in logarithmic scale. When X_{lin} equals the reference level, the dB-scale becomes zero:

X_{dB}=10\log_{10}\left.\left(\frac{X_{lin}}{X_{ref}}\right)\right|_{X_{lin}=X_{ref}}=0dB.

Furthermore, when X_{lin}>X_{ref}, X_{dB} is positive and if X_{lin}

So far, so good. But, on the other hand, more often than not, you also see the following definition:

Y_{dB}=20\log_{10}\left(\frac{Y}{Y_{ref}}\right),

i.e. the factor before the logarithm is 20 instead of 10. Which version is correct? Both formulas are correct, but ...

Python OFDM Example

Maximilian Matthe — Wed, 01 Feb 2017 00:00:00 +0100

Basic OFDM Example in Python¶

In this notebook, we will investigate the basic building blocks of an OFDM system at the transmitter and receiver side. OFDM (Orthogonal frequency division multiplexing) is a multicarrier system that is applied in a wide range of wireless transmission systems, such as LTE, WiMAX and DVB-T and DAB. The fundamental concept of a multicarrier system is the division of a high-rate transmitted data stream into several low-rate narrow subcarriers. This way, several advantages are obtained:

Since the symbol duration is inverse proportional to the symbol rate, each subcarrier has relatively long symbols. Long symbols are robust against multipath fading, as it occurs in wireless systems.
When a carrier is in a deep fade due to frequency-selectivity of the channel (i.e. the received energy on this carrier is very low), only the data on this subcarrier is lost, instead of the whole stream.
Multicarrier systems allow easy multi-user resource sharing by allocating different subcarriers to different users.

Consider the following block diagram, which contains fundamental blocks for the OFDM system:

In the following OFDM ...

Eye diagram examples

Maximilian Matthe — Sun, 22 Jan 2017 00:00:00 +0100

The Eye Diagram¶

In this notebook, we are going to draw eye diagrams of different pulse shaping filters and evaluate, if they fulfill the first and second Nyquist criterion.

Let g(t) be some pulse shaping filter which is used to pulse-shape transmit symbols in the baseband. When the (complex-valued) data symbols are given by d[k], the overall transmit signal x(t) is given by

x(t)=\sum_{k\in\mathbb{Z}}d[k]g(t-kT),

where 1/T is the symbol rate and T is the time distance between adjacent symbols. We can understand the transmit signal as the superposition of time-shifted pulse shaping filters that are multiplied with the data-symbol. This technique is common to most communication systems.

Let us first define a function get_filter which returns different kinds of pulse shaping filters, depending on the parameters.

def get_filter(name, T, rolloff=None):
    def rc(t, beta):
        import warnings
        with warnings.catch_warnings():
     ...

Spectral Leakage, Zero-Padding and Frequency Resolution

Maximilian Matthe — Sat, 21 Jan 2017 00:00:00 +0100

Spectral Leakage and Zero-Padding of the Discrete Fourier Transform¶

The discrete Fourier Transform is of extreme importance for all kinds of digital signal processing tasks. Given an input sequence x[n] of length N samples, the DFT is given by X[k] = \sum_{n=0}^{N-1}x[n]\exp(-j2\pi\frac{nk}{N}). It is used for filtering, spectral analysis, complexity reduction, equalization and so many other kinds of analysis. In this article, we are going to illustrate the properties of the discrete Fourier Transform which are different from the conventional, time-continuous Fourier transform. We will describe the effect of zero-padding versus using a larger FFT window for spectral analysis. Finally, we will apply this knowledge to the detection of dual-tone multi-frequency signaling, how it is used in the standard phone dialing.

Let us first pose the central property of the DFT:

The Discrete Fourier Transform (DFT) assumes that its input signal is one period of a periodic signal. Its output are the discrete frequencies of this periodic signal.

Fourier Transform of periodic and windowed functions¶

This central property is very important to understand. Let us first examine the (continuous-time) Fourier transform of some periodic signal. To do this, we define a function that calculates the continuous-time Fourier Transform (see also this post ...

The Dirac Comb and its Fourier Transform

Maximilian Matthe — Wed, 18 Jan 2017 00:00:00 +0100

The tikzmagic extension is already loaded. To reload it, use: %reload_ext tikzmagic

The Dirac Comb function¶

The continuous-time comb function C_T(t) is an important tool in signal processing and sampling theory and given by

C_T(t)=T\sum_{n=-\infty}^{\infty}\delta(t-nT).

Let us first draw this Dirac comb function. Due to the infinitely small width of the Dirac impulse, it is tough to numerically model the Dirac comb in continuous time. Let us therefore resort to discrete time for the visualization:

T = 2                      # time distance between pulses
Fs = 1000                  # sampling frequency, used for discretizing the system
t = np.arange(-6, 6, 1/Fs) # time range to consider
comb = np.zeros_like(t)
comb[::int ...

Approximating the Fourier Transform with DFT

Maximilian Matthe — Sun, 08 Jan 2017 00:00:00 +0100

Using the Discrete Fourier Transform to approximate the Continuous-Time Fourier Transform¶

In this article, we want to understand, how we can employ the discrete Fourier transform to approximate the continuous-time Fourier Transform. We are particularly interested, because there exist fast implementations of the DFT (see Fast-Fourier-Transform which make the calculations much quicker and easier to accomplish. Furthermore, in the world of digital signal processing, most signals are discrete, making the DFT a very suitable tool.

Let us remember the Fourier Integral for a time-limited function g(t) with g(t)=0, |t|>t_0. It is given by

G(f)=\int_{-t_0}^{t_0} g(t) \exp(-j2\pi ft)dt,

where the integration limits are chosen because g(t)=0 outside of this interval. In the following, we will derive how this integral can be approximated by the discrete Fourier transform.

To confirm our results, let us first define the Python function cft, which numerically evaluates this integral by means of the Simpsons Integration rule:

def cft(g, f):
    """Numerically evaluate the Fourier Transform of g for the given frequencies"""    
    result = np.zeros ...

Properties of the Fourier Transform

Maximilian Matthe — Sun, 08 Jan 2017 00:00:00 +0100

Properties of the Fourier Transform¶

In this article, we will illustrate several basic properties of the Fourier Transform, which are essential for working with the transform from day to day. First, let us define a Python function which approximates the Fourier transform

X(f) = \int_{-\infty}^{\infty}x(t)\exp(-j2\pi ft) dt

from a sampled version x(\frac{n}{F_s}-t_0), n=0,\dots,N-1 of the signal, by using the discrete Fourier transform (refer to Approximation of Fourier Transform with DFT for an explanation):

def ft(samples, Fs, t0):
    """Approximate the Fourier Transform of a time-limited signal 
    by means of the discrete Fourier Transform.
    
    samples: signal values sampled at the positions t0 + n/Fs
    Fs: Sampling frequency of the signal
    t0: starting time of the sampling of the signal
    """
    f = np.linspace(-Fs/2, Fs/2, len(samples), endpoint ...

The first Nyquist criterion

Maximilian Matthe — Sun, 08 Jan 2017 00:00:00 +0100

The first Nyquist Criterion¶

{% include "references/digitalmodulationscourse.html" %}

This article describes the 1st Nyquist Criterion, which is of central importance in the design of filters that are used for digital data transmissions. We provide expressions for the time-domain and frequency-domain and illustrate the results with Python source code.

Let us first define a python function ft which calculates the Fourier Transform of samples of a time-continuous function (refer to Approximation of the Fourier Transform). Also, define a function get_filter that can return different kinds of pulse shaping filters, which we use throughout this notebook.

def ft(samples, Fs, t0):
    f = np.linspace(-Fs/2, Fs/2, len(samples), endpoint=False)
    return np.fft.fftshift(np.fft.fft(samples)/Fs * np.exp(-2j*np.pi*f* ...

Welcome to DSPIllustrations.com!

Maximilian Matthe — Sun, 01 Jan 2017 00:00:00 +0100

Welcome to DSPIllustrations.com!

I believe that digital signal processing and fundamentals of communications systems dont need to be taught on purely theoretic and mathematical level. There are so many things that can be easily visualized instead of being derived from tedious calculations.

This site is dedicated to illustrate fundamental aspects of signal processing and analysis with easy to follow code examples and graphical illustrations. This mixture of basic mathematical theory and direct illustration with source code allows to

better understand the treatment,
ensure correctness by numerical verification,
learn, how equations can be modeled in computer programs.

Throughout the site, I use the Python programming language. Even though MATLAB is more popular among communication engineers, Python has the following particular advantages:

Python is free. You dont need to buy an expensive license, be connected to a universities license server or go the risky way of downloading a cracked version.
Python is a general-purpose language, which has dedicated libraries to support signal processing. Hence, it is easier to structure your program and you're not lost, when you need to perform tasks that are not related to signal processing itself.
Python supports literal programming with the use of Jupyter Notebooks. ...