Dirac Delta Function – δ(t)

Introduction

The unit impulse or Dirac Delta function δ(t) is an essential building block of modern telecommunications, similar in importance to the FFT that was examined in the previous post. The function came about as a result of research done by the British Physicist Paul Dirac on the modelling of a point charge (Ref.1). It is actually a so called distribution or a generalized function, and its properties can be examined in the limiting sense (Ref.2 p45). These can be visualized very effectively using telecom software such as Scicos or GNU Radio.

Properties

The essential properties can be summarized by the following equations shown in Figure 1. The unit impulse is zero everywhere except at the origin where it has an infinite amplitude with an infinitely small duration such that the area remains 1. The Fourier Transform yields an infinite spectrum with unit value. In the real world this means that a very fast pulse, similar to a radar pulse, has a very wide flat spectrum.

Figure 1 Properties of Unit Impulse

Limiting Behaviour

The unit impulse or Dirac Delta function δ(t) can be modeled by considering the Fourier Series of a Rectangular Pulse Train as shown in Figure 2. The unit impulse is the pulse train with an amplitude A = [1/duration] so that the area = A x duration = 1 and with T —> infinity & duration —> zero.

Figure 2 Fourier Series Expansion of Rectangular Pulse Train

Scicos Model Rectangular Pulse Train

Figure 3 shows a Scicos model that can be used to examine the unit impulse. We start off with an RPT Rectangular Pulse Train waveform of duration = 10msec and period of 100msec. Since this is a periodic waveform, it has Fourier components that are spaced by 1/period = 10Hz and spectral nulls at 1/pulse_width = 1/10msec = 100Hz. This is shown in Figure 4. If we now decrease the pulse width to 5%, the null moves out to 200Hz, and the components are still spaced by 10Hz. This is shown in Figure 5. In Figure 6, we increase the period to 200msec. Now the components are only spaced by 5Hz. This is shown in Figure 6.

The Unit Impulse is the RPT where the period goes to infinity, thus making the spacing between components infinitely small or a continuous spectrum. The amplitude goes to infinity and the duration goes to zero keeping the area =1. Since the duration goes to zero, the spectral null moves out to infinity. Thus we have the Fourier transform in the limit.

Figure 3 Scicos Model of RPT Rectangular Pulse Train
Figure 4 RPT d=10%=10msec/T=100msec
Figure 5 RPT d=5%=5msec/T=100msec
Figure 6 RPT d=10%=20msec/T=200msec

References

#1 – “Dirac Delta Function”, Wikipedia

#2 – “Communication Systems: An Introduction to Signals and Noise in Electrical Communication”, A. Bruce Carlson, McGraw-Hill, 1968, Library of Congress 68-2276109955


FFT – A Telecommunications Revolution

The FFT or Fast Fourier Transform was described by Cooley & Tukey in their 1965 paper “An Algorithm For The Machine Calculation of Complex Fourier Series” (Ref.1). The method has earlier roots in the works of Carl Gauss and others (Ref.2). The purpose of the FFT is to reduce the calculation complexity of the DFT or Discrete Fourier Transform.

On a practical level, I remember my first exposure to a Spectrum Analyzer in 1976 was the venerable HP141T. This was an amazing instrument that had various plug-ins depending on the band of interest. Only experienced senior engineers were allowed to use it. It had two hefty handles on the front and had a special carrying case that weighed a ton when loaded. Fast forward to 2018 and I now use lightweight portable usb powered instruments that plug into my laptop. All of this is due to the digital revolution and in particular the FFT that makes all of this possible.

HP141T Spectrum Analyzer
USB Portable Spectrum Analyzers

References

#1. “An Algorithm For The Machine Calculation of Complex Fourier Series”, James W. Cooley, John W. Tukey, Journal Mathematics of Computation #19, 1965.

#2. “Fast Fourier Transform”, Wikipedia

GNU Radio ATSC_8VSB Transmitter Simulation

Figure 1 CBLFT UHF CH25

In the previous post, I discussed using Splat! to predict the receive level of an RF signal. In particular I considered CBLFT UHF CH25 as received from the CN Tower as shown above in Figure 1. In order to study ATSC_8VSB which is the DTV standard used in North America, GNU Radio can be used to actually simulate a transmitter and receiver.

Figure 2 ATSC DTV Standard Signal Flow

Figure 2 shows the block diagram of ATSC_8VSB from Ref.1. Video information is compressed by an MPEG-2 video coder and audio is compressed by a Dolby AC-3 coder. These two streams are mixed with Auxiliary & Control data to form a MPEG-2 transport stream of packets. These packets undergo Channel Coding and 8VSB Modulation with modules as shown in Figure 3.

Figure 3 ATSC Channel Coding & Modulation Modules

Figure 4 shows a GNU Radio simulation of ATSC_8VSB transmission. An MPEG-2 video file is read into the simulation and processed by the channel coding blocks. The signal is then 8VSB modulated to UHF CH25 and the spectrum displayed.

Figure 4 UHF CH25 ATSC_8VSB Simulation with Video & Spectrum

References:

#1 – “A/53 ATSC Digital Television Standard Parts 1 – 6, 2007
https://www.atsc.org/standard/a53-atsc-digital-television-standard

#2 – “GNU Radio 3.7.0 Documentation – gnuradio.atsc Signal Blocks”
https://www.gnuradio.org/doc/sphinx-3.7.0/atsc/index.html

Splat! RF Signal Propagation, Loss and Terrain Analysis Tool

About Splat!

Splat! is an RF Signal Propagation, Loss, And Terrain analysis tool for the electromagnetic spectrum between 20 MHz and 20 GHz. Splat! software is Copyright © 1997-2018 by John Magliacane KD2BD. Ref.1.

Splat! is free software. It may be redistributed and/or modified under the terms of the GNU General Public License Version 2 as published by the Free Software Foundation.

Properties:
*Terrestrial signal propagation & terrain analysis tool
*Frequency range 20MHz – 20GHz (VHF, UHF, Microwave)
*Antenna height determination
*Fresnel Zone clearance requirements
*Path Loss
*Received signal level
*Longley-Rice or ITM Irregular Terrain Model
*Signal level contour maps for coverage
*Originally developed for 1997 for College ATV repeater
*Longley Rice developed for TV propagation study

Theory

Propagation > 30MHz works by various mechanisms: Line of Sight LOS, diffraction over the horizon, atmospheric refraction and atmospheric scatter.

Linux Installation

Download and install Ubuntu 18.04.1 natively or on VMware Workstation Player 14 (Virtual Machine on Windows). Once Ubuntu is installed, ensure everything is updated. The Linux version of Splat is a command line interface Ref.1, documentation is available Ref.2.

sudo apt-get update
sudo apt-get install splat
$splat

Windows Installation

Download and install Splat with Windows GUI – Beta v1.1.2 from Austin Wright’s web site Ref.3. Don’t install in the C:/Program Files directory, install directly in C:/SPLAT. Note that the Windows version of Splat has a visual interface.

Elevation Data Format

Splat! requires DEM Digital Elevation Data in order to perform the various propagation calculations. The data must be in the form of SDF Splat Data Files .sdf. The digital elevation data may be in several formats depending on the source:

*GeoTIFF (.tif)
*USGS DEM (.dem)
*Floating Point Raster File (.flt)
*ASCII Grid (.txt)
*Comma Separated Values (.csv or .txt)
*SRTM (.hgt)

Splat! has several utilities to convert .dem and .hgt files to the native Splat! .sdf format. If the data is not in this format, there are several Open Source programs that can be used to convert the data to .dem/.hgt such as GDAL Ref.4 & QGIS Ref.5.

Canadian DEM Files

Figure 1

Canadian DEM files are available from Open Canada Data Ref.6. Figure 1 shows an index for Canadian DEM files. They are 1:250,000 scale in GeoTIFF format. They are based on NAD 83, resolution 0.75arc_sec in N-S, 0.75-3arc_sec in W-E.

US DEM Files

DEM files in the US are available from NED National Elevation Data Set Ref.7. The data set is a raster product with seamless matching. The following resolutions are possible:

*1/3arc_sec (10m) for US & Alaska
*1arc_sec (30m) for US, Hawaii, Puerto Rico, Canada, Mexico
*NAD83 and 1m elevation unit

Earlier DEM files with various formats are available from Earth Explorer Ref.8. Note that 1 minute of arc is defined as 1 Nautical Mile:

*1 minute arc = 1 Nautical Mile = 1852m
*1 second of arc = 1852/60 = 30.87m or approx. 31m

In order to download SRTM data from USGS, which is immediately useful for Splat!, you need to register and setup an account. Once you submit the details a confirmation email will be sent to you. Figure 2 shows Earth Explorer giving details of SRTM data around Toronto. A search box was created around the Toronto area and data type of digital elevation/srtm used.

Figure 2

Splat! Data Utilities

Splat! requires DEM data in the native Splat! Data Format .sdf. There are two utilities that can be used to convert SRTM .hgt format to the .sdf format.

srtm2sdf.exe         converts 3arc_sec .hgt files to .sdf
srtm2sdf-hd.exe   converts 1arc_sec .hgt files to .sdf

Newer SRTM data from Earth Explorer is now in GeoTIFF format, so another intermediate Utility GDAL is required to convert the GeoTIFF .tif to .hgt first.

Splat! QTH Files

In order to study the propagation between two locations in the VHF, UHF and Microwave frequency bands, we have to consider the influence of terrain between the transmitter and the receiver. To do this, we first need to know the location and height of the Tx/Rx antennas. In Splat! we do this by creating QTH files. Care must be taken entering Latitude and Longitude. Antenna height is specified in feet (default) or in meters if ‘meters’ is added. Note Longitude entry is opposite to Google Earth.

*Latitude    = + (0/90)deg for North Latitude
*Latitude    = – (0/90)deg for South Latitude
*Longitude = + (0/360)deg for West Latitude
*Longitude = – (0/360)deg for East Longitude

Figure 3

Figure 3 shows a QTH file for the CBLFT transmitter located in the CN Tower Toronto. A similar file is constructed for the receive location VE3PKC.

Figure 4

Figure 4 shows the path 7.56Km between the CBLFT UHF CH25 transmitter located in the CN Tower to the VE3PKC location. The figure shows the terrain immediately under the path, antenna heights and 100% 1st Fresnel zone clearance. The Fresnel Zone Ref.9 is an ellipsoid of revolution about the direct line joining the Tx/Rx antennas. For good signal strength, this regions should be at least 60% free of any obstacles. The nth Fresnel zone is defined as the locus of points in space such that the two segment path reflection off the surface is (n/2) x wavelength longer than the direct path. This results in wave cancellation when the two waves add.

Propagation Calculation CBLFT to VE3PKC

We can use the Friis formula Ref. 10 to estimate the receive signal level of CBC French TV station CBLFT transmitting from the CN Tower to the VE3PKC location. Figure 5 shows the results.

ERP = 106.2KW ERP (Tx Power + Tx Ant Gain)
Rx = Receive Signal Power in dBm
Tx = Transmit Power in dBm = +80.26dbm
Gtx, Grx = Antenna Gains in dBi = +2.2dB
Lrx = Receive transmission Line Loss dB = 2.6dB
d=7.56Km
f=536.31MHz
FSPL = Free Space Path Loss = 32.44 + 20log10(d_Km) + 20log10(f_MHz) = 104.6dB
Rx = Tx + Gtx -FSPL -Lrx + Grx = -22.5dBm

Figure 5

Assuming an ERP of 106.2KW (EIRP = +2.2dB), the receive power at VE3PKC is -22.5dBm with a receive antenna gain of 2.2dBi & receiver feeder loss of 2.6dB (half wave dipole cut 536.6MHz). Figure 6 shows a 6MHz receive channel power of -35.8dBm. This corresponds to an extra 13.3dB of attenuation. Several factors have to be considered here. The full antenna EIRP may not be directed at the VE3PKC location and thirdly, although no objects are in the Fresnel zone under the path, we can see that the signal path goes through a tunnel of buildings either side of the path, which may be in the Fresnel zone(s).

Figure 6

Splat! Propagation Calculation CBLFT to VE3PKC

We can use Splat! to calculate the propagation loss also taking into account any Fresnel zone loses. In order to do this we need to create an .lrp file for the CN Tower transmitter location. Figure 7 shows the entry menu. The Splat! manual Ref.2/p3 can be used to help filling in these values. Transmitter power ERP is the last field.

Figure 7

The Splat! results are the same as the Friis calculations. Free space attenuation is 104.6dB and receive power is -22.1dBm before Rx antenna gain or feeder loss giving -22.5dBm at the receiver input after antenna/feeder.

Figure 8

References

#1 – Splat Home Page
http://www.qsl.net/kd2bd/splat.html

#2 – Splat Documentation
http://www.qsl.net/kd2bd/splat.pdf

#3 – Splat with Windows GUI Version – Beta 1.1.2, Austin Wright
http://www.ve3ncq.ca/wordpress/?page_id=62

#4 – GDAL Geospatial Data Abstraction Layer
https://www.gdal.org

#5 – QGIS Open Source Geographic Information System
http://www.qgis.com

#6 – Canadian Elevation Data
https://open.canada.ca/data/en/dataset/7f245e4d-76c2-4caa-951a-45d1d2051333

#7 – NED National Elevation Data Set
https://lta.cr.usgs.gov/NED

#8 – Earth Explorer
https://earthexplorer.usgs.gov

#9 – Fresnel Zone
https://en.wikipedia.org/wiki/Fresnel_zone

#10 – Friis Transmission Equation
https://en.wikipedia.org/wiki/Friis_transmission_equation

#11 – “Splat!: An RF Signal Propagation, Loss and Terrain Analysis Tool”, John A. Magliacane KD2BD, Bill Walker W5GFE, QEX Magazine, July/August 2009
http://www.arrl.org/news/i-qex-i-the-july-august-2009-issue

Balcony ATSC HD Television

For some time I have wanted to see how many television channels I could receive from the CN Tower in Toronto. I can clearly see the TV antenna at the top of the tower from my apartment balcony, although it is boxed in on both sides by buildings. The 7.6Km Google Earth path is shown below.

I used a simple Yagi antenna and mounted it on a wooden dowel combination to easily allow it be rotated and temporarily but securely attached to the apartment balcony railing.

uhf_tv_ant
uhf_tv_ant

I placed my TV in scan mode to see what it would pickup. I was amazed to find that I detected 24 channels listed in the spread sheet below:

tvstn_cntwr
tvstn_cntwr

I also scanned from 400MHz to 800MHz with my spectrum analyzer to see what I would detect:

A close up on the strongest signal at 536.3MHz shows the following spectrum with carrier level at -60dBm and spectrum at -73dBm:

RTL-SDR for Weather Radio

rtlsdr_wx
rtlsdr_wx

A simple RTL-SDR receiver can be used to receive marine weather forecasts. The unit is small and inexpensive (approx. $25). It can be controlled by using the software program SDR# or other similar programs. The picture above shows my unit plugged into the usb port on the left side of my laptop, the display shows Weather Radio Canada station XMJ-225 on 162.4MHz.  The picture below show a simple whip antenna on my apartment balcony.

whip_ant
whip_ant