Simulation of Digital Low Pass Filter and Amplitude Modulation-Demodulation in Matlab Simulink Environment

This work consists of two research section namely; simulation of digital low pass filter and amplitude modulation-demodulation in Matlab Simulink environment. The purpose of this experiment is to investigate the characteristics of the digital low pass filter under different frequencies. In addition, the section on amplitude modulation-demodulation was conducted to investigate similarities and differences between product detectors and envelop detector as modes of demodulation. Furthermore, the spectrum analysis of modulated and demodulated signals was investigated.

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Abstract.
This work consists of two research section namely; simulation of digital low pass filter and
amplitude modulation-demodulation in Matlab Simulink environment. The purpose of this
experiment is to investigate the characteristics of the digital low pass filter under different
frequencies. In addition, the section on amplitude modulation-demodulation was conducted to
investigate similarities and differences between product detectors and envelop detector as
modes of demodulation. Furthermore, the spectrum analysis of modulated and demodulated
signals was investigated.

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Introduction
A low-pass filter is a circuit that enables frequencies of the range 0 - f_c Hz (zero to cutoff
frequency) to pass and blocks or attenuates frequencies greater than the cutoff frequency. Digital
processors are embedded in digital filters to compute algorithm on signal’s sampled values.
Superiority of digital filters over analog-type filters is demonstrated by the fact that they have higher
accuracy and ability to restore distorted signals (Smith).
Modulation is the act of translating a message signal into higher frequency signal so as to
enable simultaneous transmission of more baseband signals when translated at different frequencies,
and also to effectively minimize the size of higher frequency reception antenna at the receiver
(Spectrum Analysis). Demodulation takes place at the receiver end where the original message is
recovered from the carrier signal.
Results and analysis
a) Low-pass digital filter
When the sampling time is 0.001 seconds and the samples in one period is 256, then analog
frequency of the sine wave was calculated as:
Period of the analog signal ; T =( samples
period × ts )
T =256 × 0.001 s=0.256 s
The frequency of the analog signal, f t
f t= 1
T = 1
0.256 =3.906 Hz
Graphical view of the input signal and the output signal when at 256 samples within the
signal’s period is as shown in fig 1.1:

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Fig1.1 The Input waveform and output signals at 256 (samples/ period)
Fig 1.2 The Input waveform and output waveform at 4 samples per period
Setting samples at 256 per period, from (Fig 1.1) the output signal can be described to be
equivalent to the input signal in all property aspects. The signals have a frequency of 3.906 Hz as
from signals’ periodic time (0.256sec). Therefore, it can be concluded that this frequency falls within
the bandwidth frequency of the designed low pass filter. A low-pass filter is permeable to low-
frequency signals and impermeable to high-frequency signals (Low pass filters).
At the sampling rate set at 4, the output signal as shown in (fig 1.2) has been suppressed to a
near-zero datum line. The frequency of the output signal is 250Hz. With regard to properties of the

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low pass filter, this frequency can be suggested to be located outside the bandwidth. Signal
components above specified frequency are attenuated by low pass filter (Carter and Mancini).
The function that gives gain at every frequency is known as the amplitude response (Smith, J). The
frequency response, amplitude against frequency, of the low-pass filter was plotted as shown below in
Matlab using the code in the appendix:
Fig 1.3. The frequency response of the low pass filter.
Addition of the second sine wave source and summation to the original sine wave whose initial
samples been changed to 64, the resultant waveforms are shown in fig 1.4.
Fig 1.4 Graphical view of the input signal, summed signal and output signal

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Summation of the high frequency (4 samples per period) signal with the low-frequency signal (64
samples per period) resulted in the original signal being chopped at the sampling rate of the high-
frequency component. The output shows a slightly distorted version of the original signal and a
suppressed higher frequency component.
b) Amplitude modulation and demodulation
Signals observed from the scope when the model was run are as shown below.
Fig 1.5 The combined signals of amplitude modulation and demodulation
D.C offset set to three
The modulated and message signals were extracted as shown below in fig 1.6.
Fig 1.6 Modulated carrier waveform and message signal

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From the observation, as evidenced in Fig 1.6, the amplitude property of the carrier signal has
been changed proportionally with the amplitude of the message signal. With an offset of 3, the
position of the message has shifted perfectly fitting onto the upper sideband of the amplitude
modulated signal. The upper sideband can be observed as having the same amplitude as the message
signal. It can be deduced that the complex modulated signal is made up of the sum of three signal
components at different frequencies. The upper sideband is a composite frequencies summation of the
carrier and message signal while the lower sideband is made of the difference between the carrier
frequency and modulation frequency.
The modulated signal was demodulated using the product detector and envelope detector
resulting in two waveforms of the original signal as shown in Fig 1.7
Fig 1.7 Demodulated signals of the product (yellow waveform) and envelope detector (blue)
For both types of demodulation, the average output of the demodulated signals is attenuated
equivalent of the message signal. Apparently, these two waves are in phase. The detectors remove
frequencies other than predefined message signal using digital means to obtain the natural frequency
of the original message signal (Detection of Amplitude Modulated Wave). However, the output signal
of the envelop detector has been attenuated with greater factor compared to the output signal of the
product detector.
Spectral analysis
1) Spectrum at amplitude modulated signal
The spectrum is as shown in the fig 1.8.

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Fig 1.8 The spectrum of AM signal
The spectrum has combination three frequency components at 0.9kHz, 1kHz and 1.1kHz. The
dominant peak is at 1kHz while the 0.9kHz and 1.1kHz are the lower and upper side bands
respectively.
2) Connected at AM signal after the oscillator before LPF
The spectrum is as shown below:
Fig 1.9 The spectrum of AM signal in the product detector.
As from the figure above, it can be observed that the spectrum consists of four frequency components
at 0.1kHz, 1.9kHz, 2kHz and 2.1kHz. Seemingly, the horizontal position of the dominant peak has
shifted to the right with respect to the previous spectrum.

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3) The spectrum of the demodulated signal
The spectrum display is as shown below:
Fig 1.10 The spectrum of the demodulated signal
The spectrum is made of one frequency component whose dominant peak is at 100Hz. This is
the pure sine wave of the recovered message signal after the high-frequency component of the carrier
signal has been attenuated by the low-pass filter.
D.C offset set to zero
The amplitude of the modulated signal with reference to the message signal:
Fig 1.11 The modulated and message signal at zero offset
The message signal no longer follows any sideband of the modulated signal. The signal is
overmodulated since the carrier is cut off at the modulation minima.

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Fig 1.12 The demodulated signal of the envelop detector (blue) and Product detector
(yellow)
Scrutinizing both waves meticulously, it was observed that these signals are not the
equivalent replica of the message signal. They have different frequencies and amplitude.
Spectral analysis at offset=0
1) Spectrum at amplitude modulated signal
Fig 1.13 The spectrum of the AM signal at zero offset
The signal has two frequency components with equivalent peaks at 0.9kHz and 1.1kHz.
There is no dominant peak as the side bands do not form pure sine waves profiles of the
message signal.

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2) Connected at AM signal after the oscillator before LPF
Fig 1.14 The spectrum of the AM signal in the Product detector
The spectrum has three frequency components at 0.1kHz, 1.9kHz and 2.1kHz. The dominant
peak is not located within the sidebands.
3) Spectrum of the demodulated signal
Fig 1.15 The spectrum of the product detector output
The spectrum has one frequency component at 100Hz at the output of the product detector. The
distorted signal has been reconstructed.

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Further analysis was done by connecting the spectrum analyzer to the output of the envelope detector.
Fig 1.16 The spectrum of the envelop detector output
The demodulated signal had three frequency components showing distortion in the message signal
resulting in the output of product and envelope detector having their outputs at different frequencies.
Conclusion
A digital filter was implemented and simulated in Matlab Simulink. The number of samples
per period was found to be inversely proportional to the frequency of the resultant signal. At higher
sampling per period, the resultant waves had relatively low frequencies. At these frequencies, signals
passed through the low pass filter. Inversely, at low sampling per period, the resultant signals had
relatively higher frequencies thus were attenuated or blocked by low pass filters. From the frequency
response of low pass filter, the cut-off frequency at 3dB was approximated to 112Hz.
In amplitude modulation and demodulation, the signals were studied at stages through the
modulation-demodulation channels. The comparison was made between product detectors and
envelop detector. Resultant sine waves were further studied in their frequency domain whose
spectrums were analyzed. It was found that the product detector is more efficient as compared to the
envelop detector because its demodulated signal was less attenuated. Also, when over-modulated

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wave at zero offset was injected into the design, the product detector was able to reconstruct and
recover the distorted message signal.
Appendix
clear all
%LOW PASS FILTER FREQUENCY RESPONSE
%Analogy frequency calculated using the formulae ft=1/T
Freq=[3.9 4.1 4.3 4.5 4.8 5.1 5.4 5.8 6.2 6.8 7.4 8.1 8.9 10 11.4 13.2 15.6 19.2 25 35.7 62.5 250];
%Value of the amplitudes as obtained from simulink
Amp=[0.9796 0.9795 0.9795 0.9794 0.9794 0.9793 0.9792 0.9791 0.9789 0.9787 0.9785 0.9782 0.9778 0.9772
0.9764 0.9752 0.9732 0.9697 0.9626 0.9447 0.8747 0.2300];
%Plotting frequency response of the Low Pass Filter
plot(Freq,Amp)
title('FREQUENCY RESPONSE OF LOW PASS FILTER')
xlabel('Freq (Hz)')
ylabel('Amplitude')
grid on

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Works cited
Bruce Carter and Ron Mancini. “Active Filter Design Techniques”. Op Amps for Everyone.
5th ed.Elsevier.2018
Detection of Amplitude Modulated Wave. Toppr. 2017.
<https://www.toppr.com/guides/physics/communication-systems/detection-of-
amplitude-modulated-wave/>
Low pass filters. All About Circuits. EETech.
<https://www.allaboutcircuits.com/textbook/alternating-current/chpt-8/low-pass-
filters/>
Smith, Steven. “Introduction to Digital Filters.” The Scientist and Engineer’s Guide to
Digital Signal Processing. Carlifornia Technical Publishing. 2011.
https://www.dspguide.com/ch14/1.htm
Smith, Julius. “Frequency Response.” Introduction to Digital Filters. CCRMA.2007
<https://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_I.html >

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Spectrum Analysis Amplitude and Frequency Modulation. Keysight Technologies. 2014.
<https://literature.cdn.keysight.com/litweb/pdf/5954-9130.pdf>