Tag: TFS

Phase demodulation via Hilbert transform: Hands-on

Key focus: Demodulation of phase modulated signal by extracting instantaneous phase can be done using Hilbert transform. Hands-on demo in Python & Matlab.

Phase modulated signal:

The concept of instantaneous amplitude/phase/frequency are fundamental to information communication and appears in many signal processing application. We know that a monochromatic signal of form x(t) = a cos(ω t + ɸ) cannot carry any information. To carry information, the signal need to be modulated. Different types of modulations can be performed – amplitude modulation, phase modulation / frequency modulation.

In amplitude modulation, the information is encoded as variations in the amplitude of a carrier signal. Demodulation of an amplitude modulated signal, involves extraction of envelope of the modulated signal. This was discussed and demonstrated here.

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
● Wireless communication systems in Matlab ISBN: 979-8648350779
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In phase modulation, the information is encoded as variations in the phase of the carrier signal. In its generic form, a phase modulated signal is expressed as an information-bearing sinusoidal signal modulating another sinusoidal carrier signal

\[x(t) = A cos \left[ 2 \pi f_c t + \beta + \alpha sin \left( 2 \pi f_m t + \theta \right) \right]   \quad \quad \quad (1)\]

where, m(t) = α sin (2 π fm t + θ ) represents the information-bearing modulating signal, with the following parameters

α – amplitude of the modulating sinusoidal signal
fm – frequency of the modulating sinusoidal signal
θ – phase offset of the modulating sinusoidal signal

The carrier signal has the following parameters

A – amplitude of the carrier
fc – frequency of the carrier and fc>>fm
β – phase offset of the carrier

Demodulating a phase modulated signal:

The phase modulated signal shown in equation (1), can be simply expressed as

\[x(t) = A cos \left[ \phi(t) \right]    \quad\quad\quad (2) \]

Here,  ɸ(t) is the instantaneous phase  that varies according to the information signal m(t).

A phase modulated signal of form x(t) can be demodulated by forming an analytic signal by applying Hilbert transform and then extracting the instantaneous phase. This method is explained here.

We note that the instantaneous phase is ɸ(t) = 2 π fc t + β + α sin (2 π fm t + θ) is linear in time, that is proportional to 2 π fc t . This linear offset needs to be subtracted from the instantaneous phase to obtained the information bearing modulated signal. If the carrier frequency is known at the receiver, this can be done easily. If not, the carrier frequency term 2 π fc t needs to be estimated using a linear fit of the unwrapped instantaneous phase. The following Matlab and Python codes demonstrate all these methods.

Matlab code

%Demonstrate simple Phase Demodulation using Hilbert transform
clearvars; clc;
fc = 240; %carrier frequency
fm = 10; %frequency of modulating signal
alpha = 1; %amplitude of modulating signal
theta = pi/4; %phase offset of modulating signal
beta = pi/5; %constant carrier phase offset 
receiverKnowsCarrier= 'False'; %If receiver knows the carrier frequency & phase offset

fs = 8*fc; %sampling frequency
duration = 0.5; %duration of the signal
t = 0:1/fs:1-1/fs; %time base

%Phase Modulation
m_t = alpha*sin(2*pi*fm*t + theta); %modulating signal
x = cos(2*pi*fc*t + beta + m_t ); %modulated signal

figure(); subplot(2,1,1)
plot(t,m_t) %plot modulating signal
title('Modulating signal'); xlabel('t'); ylabel('m(t)')

subplot(2,1,2)
plot(t,x) %plot modulated signal
title('Modulated signal'); xlabel('t');ylabel('x(t)')

%Add AWGN noise to the transmitted signal
nMean = 0; %noise mean
nSigma = 0.1; %noise sigma
n = nMean + nSigma*randn(size(t)); %awgn noise
r = x + n;  %noisy received signal

%Demodulation of the noisy Phase Modulated signal
z= hilbert(r); %form the analytical signal from the received vector
inst_phase = unwrap(angle(z)); %instaneous phase

%If receiver don't know the carrier, estimate the subtraction term
if strcmpi(receiverKnowsCarrier,'True')
    offsetTerm = 2*pi*fc*t+beta; %if carrier frequency & phase offset is known
else
    p = polyfit(t,inst_phase,1); %linearly fit the instaneous phase
    estimated = polyval(p,t); %re-evaluate the offset term using the fitted values
    offsetTerm = estimated;
end
    
demodulated = inst_phase - offsetTerm;

figure()
plot(t,demodulated); %demodulated signal
title('Demodulated signal'); xlabel('n'); ylabel('\hat{m(t)}');

Python code

import numpy as np
from scipy.signal import hilbert
import matplotlib.pyplot as plt
PI = np.pi

fc = 240 #carrier frequency
fm = 10 #frequency of modulating signal
alpha = 1 #amplitude of modulating signal
theta = PI/4 #phase offset of modulating signal
beta = PI/5 #constant carrier phase offset 
receiverKnowsCarrier= False; #If receiver knows the carrier frequency & phase offset

fs = 8*fc #sampling frequency
duration = 0.5 #duration of the signal
t = np.arange(int(fs*duration)) / fs #time base

#Phase Modulation
m_t = alpha*np.sin(2*PI*fm*t + theta) #modulating signal
x = np.cos(2*PI*fc*t + beta + m_t ) #modulated signal

plt.figure()
plt.subplot(2,1,1)
plt.plot(t,m_t) #plot modulating signal
plt.title('Modulating signal')
plt.xlabel('t')
plt.ylabel('m(t)')
plt.subplot(2,1,2)
plt.plot(t,x) #plot modulated signal
plt.title('Modulated signal')
plt.xlabel('t')
plt.ylabel('x(t)')

#Add AWGN noise to the transmitted signal
nMean = 0 #noise mean
nSigma = 0.1 #noise sigma
n = np.random.normal(nMean, nSigma, len(t))
r = x + n  #noisy received signal

#Demodulation of the noisy Phase Modulated signal
z= hilbert(r) #form the analytical signal from the received vector
inst_phase = np.unwrap(np.angle(z))#instaneous phase

#If receiver don't know the carrier, estimate the subtraction term
if receiverKnowsCarrier:
    offsetTerm = 2*PI*fc*t+beta; #if carrier frequency & phase offset is known
else:
    p = np.poly1d(np.polyfit(t,inst_phase,1)) #linearly fit the instaneous phase
    estimated = p(t) #re-evaluate the offset term using the fitted values
    offsetTerm = estimated
                           
demodulated = inst_phase - offsetTerm 

plt.figure()
plt.plot(t,demodulated) #demodulated signal
plt.title('Demodulated signal')
plt.xlabel('n')
plt.ylabel('\hat{m(t)}')

Results

Figure 1: Phase modulation - modulating signal and modulated (transmitted) signal
Figure 1: Phase modulation – modulating signal and modulated (transmitted) signal
Figure 2: Demodulated signal from the noisy received signal

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For further reading

[1] V. Cizek, “Discrete Hilbert transform”, IEEE Transactions on Audio and Electroacoustics, Volume: 18 , Issue: 4 , December 1970.↗

Topics in this chapter

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

Books by the author


Wireless Communication Systems in Matlab
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Digital Modulations using Python
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Extract envelope, phase using Hilbert transform: Demo

Key focus: Learn how to use Hilbert transform to extract envelope, instantaneous phase and frequency from a modulated signal. Hands-on demo using Python & Matlab.

If you would like to brush-up the basics on analytic signal and how it related to Hilbert transform, you may visit article: Understanding Analytic Signal and Hilbert Transform

Introduction

The concept of instantaneous amplitude/phase/frequency are fundamental to information communication and appears in many signal processing application. We know that a monochromatic signal of form cannot carry any information. To carry information, the signal need to be modulated. Take for example the case of amplitude modulation, in which a positive real-valued signal modulates a carrier . That is, the amplitude modulation is effected by multiplying the information bearing signal with the carrier signal .

Here, is the angular frequency of the signal measured in radians/sec and is related to the temporal frequency as . The term is also called instantaneous amplitude.

Similarly, in the case of phase or frequency modulations, the concept of instantaneous phase or instantaneous frequency is required for describing the modulated signal.

Here, is the constant amplitude factor (no change in the envelope of the signal) and is the instantaneous phase which varies according to the information. The instantaneous angular frequency is expressed as the derivative of instantaneous phase.

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

Definition

Generalizing these concepts, if a signal is expressed as

  • The instantaneous amplitude or the envelope of the signal is given by
  • The instantaneous phase is given by 
  • The instantaneous angular frequency is derived as
  • The instantaneous temporal frequency is derived as

Problem statement

An amplitude modulated signal is formed by multiplying a sinusoidal information and a linear frequency chirp. The information content is expressed as and the linear frequency chirp is made to vary from to . Given the modulated signal, extract the instantaneous amplitude (envelope), instantaneous phase and the instantaneous frequency.

Solution

We note that the modulated signal is a real-valued signal. We also take note of the fact that amplitude/phase and frequency can be easily computed if the signal is expressed in complex form. Which transform should we use such that the we can convert a real signal to the complex plane without altering the required properties ?? Answer: Apply Hilbert transform and form the analytic signal on the complex plane.  Figure 1 illustrates this concept.

Figure 1: Converting a real-valued signal to complex plane using Hilbert Transform

If we express the real-valued modulated signal as an analytic signal, it is expressed in complex plane as

where, (HT[.]) represents the Hilbert Transform operation. Now, the required parameters are very easy to obtain.

The instantaneous amplitude (envelope extraction) is computed in the complex plane as

The instantaneous phase is computed in the complex plane as

The instantaneous temporal frequency is computed in the complex plane as

Once we know the instantaneous phase, the carrier can be regenerated as . The regenerated carrier is often referred as Temporal Fine Structure (TFS) in Acoustic signal processing [1].

Matlab

fs = 600; %sampling frequency in Hz
t = 0:1/fs:1-1/fs; %time base
a_t = 1.0 + 0.7 * sin(2.0*pi*3.0*t) ; %information signal
c_t = chirp(t,20,t(end),80); %chirp carrier
x = a_t .* c_t; %modulated signal

subplot(2,1,1); plot(x);hold on; %plot the modulated signal

z = hilbert(x); %form the analytical signal
inst_amplitude = abs(z); %envelope extraction
inst_phase = unwrap(angle(z));%inst phase
inst_freq = diff(inst_phase)/(2*pi)*fs;%inst frequency

%Regenerate the carrier from the instantaneous phase
regenerated_carrier = cos(inst_phase);

plot(inst_amplitude,'r'); %overlay the extracted envelope
title('Modulated signal and extracted envelope'); xlabel('n'); ylabel('x(t) and |z(t)|');
subplot(2,1,2); plot(cos(inst_phase));
title('Extracted carrier or TFS'); xlabel('n'); ylabel('cos[\omega(t)]');

Python

import numpy as np
from scipy.signal import hilbert, chirp
import matplotlib.pyplot as plt

fs = 600.0 #sampling frequency
duration = 1.0 #duration of the signal
t = np.arange(int(fs*duration)) / fs #time base

a_t =  1.0 + 0.7 * np.sin(2.0*np.pi*3.0*t)#information signal
c_t = chirp(t, 20.0, t[-1], 80) #chirp carrier
x = a_t * c_t #modulated signal

plt.subplot(2,1,1)
plt.plot(x) #plot the modulated signal

z= hilbert(x) #form the analytical signal
inst_amplitude = np.abs(z) #envelope extraction
inst_phase = np.unwrap(np.angle(z))#inst phase
inst_freq = np.diff(inst_phase)/(2*np.pi)*fs #inst frequency

#Regenerate the carrier from the instantaneous phase
regenerated_carrier = np.cos(inst_phase)

plt.plot(inst_amplitude,'r'); #overlay the extracted envelope
plt.title('Modulated signal and extracted envelope')
plt.xlabel('n')
plt.ylabel('x(t) and |z(t)|')
plt.subplot(2,1,2)
plt.plot(regenerated_carrier)
plt.title('Extracted carrier or TFS')
plt.xlabel('n')
plt.ylabel('cos[\omega(t)]')

Results

Figure 2: Amplitude modulation using a chirp signal and extraction of envelope and TFS

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Reference

[1] Moon, Il Joon, and Sung Hwa Hong. “What Is Temporal Fine Structure and Why Is It Important?” Korean Journal of Audiology 18.1 (2014): 1–7. PMC. Web. 24 Apr. 2017.↗

Topics in this chapter

Essentials of Signal Processing
● Generating standard test signals
 □ Sinusoidal signals
 □ Square wave
 □ Rectangular pulse
 □ Gaussian pulse
 □ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
 □ Real and complex DFT
 □ Fast Fourier Transform (FFT)
 □ Interpreting the FFT results
 □ FFTShift
 □ IFFTShift
● Obtaining magnitude and phase information from FFT
 □ Discrete-time domain representation
 □ Representing the signal in frequency domain using FFT
 □ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
 □ Energy of a signal
 □ Power of a signal
 □ Classification of signals
 □ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
 □ Polynomial functions
 □ Representing single variable polynomial functions
 □ Multiplication of polynomials and linear convolution
 □ Toeplitz matrix and convolution
● Methods to compute convolution
 □ Method 1: Brute-force method
 □ Method 2: Using Toeplitz matrix
 □ Method 3: Using FFT to compute convolution
 □ Miscellaneous methods
● Analytic signal and its applications
 □ Analytic signal and Fourier transform
 □ Extracting instantaneous amplitude, phase, frequency
 □ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
 □ Design specification
 □ General considerations in design

Books by the author


Wireless Communication Systems in Matlab
Second Edition(PDF)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Python
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart

Digital Modulations using Matlab
(PDF ebook)
Note: There is a rating embedded within this post, please visit this post to rate it.
Checkout Added to cart
Hand-picked Best books on Communication Engineering
Best books on Signal Processing