fm radio receiverkom.aau.dk/group/05gr506/051014/report.pdf · 2005-10-12 · a.1. purpose of the...
TRANSCRIPT
FM radio receiver
P5 projekt, AAU,Elektronik og elektroteknik
Gruppe 415Carsten
Jes Toft KristensenGustavKingo
Onkel BoyeNC
Elektronik og Elektroteknik
Fredrik Bajers Vej 7B
Telefon 96 35 98 36
Fax 98 15 36 62
http://www.esn.aau.dk
Title:
FM radio receiver
Theme:
Realtime systems
Projectperiod:P5, fall semester 2005
Project group:506
Members:CarstenJes Toft KristensenGustavKingoOnkel BoyeNC
Supervisor:Persefonis
Copies: 9
Pages: 7
Appendices: 19
Finished October 12, 2005
Synopsis:
Insert synopsis here. . . found in mainRe-
port/frontMatter/synopsis.tex
Contents
A Radiotechnolgy 7A.1 Purpose of the projekt . . . . . . . . . . . . . . . . . . . . . . . . . . 8
B Sound card 9
C Modulation 11C.1 Amplitude modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 11C.2 Frequency modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 14
D Downconverter 19D.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19D.2 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
D.2.1 Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20D.2.1.1 Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 21
D.3 Block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21D.4 Mathematical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22D.5 Dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23D.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24D.7 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
E Sampling 26E.1 From continuous to discrete . . . . . . . . . . . . . . . . . . . . . . . 26E.2 Frequency attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
F Measurements of downconverter 31F.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31F.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32F.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
G Filter 34
4
Insert text here. . .
5
BIBLIOGRAPHY
Bibliography
Christensen, Anders (1999).HF-teknik,1st edition.ISBN 87-600-0129-1(Industriens Forlag, 1999),URL www.if.dk.
Haykin, Simon (2001).Communication Systems,4th edition.ISBN 0-471-17869-1(John Wiley & Sons, Inc., 2001).
Intel Corporation (2002).Audio Codec ‘97
(Intel Corporation, 2002),URL http://www.intel.com/
design/chipsets/audio/ac97_r23.
pdf. Revision 2.3 Revision 1.0.
Johnson, David E., Johnson, Johnny R.,Hilburn, John L. & Scott, Peter D.(1999).Electric circuit analysis,3rd edition.ISBN 0-471-36571-8(John Wiley & Sons, Inc., 1999).
Laskar, Joy, Matinpour, Babak &Chakraborty, Sudipto (2004).Modern Receiver Front-Ends - Sys-
tems, Circuits, and Integration,1st edition.ISBN 0-471-22591-6(John Wiley & Sons, Inc., 2004).
Oppenheim, Alan V. & Schafer,Ronald W. (1998).Discrete time signal processing,2nd edition.ISBN 0-13-754920-2(Prentice Hall, 1998).
6
APPENDIX A. RADIOTECHNOLGY
Appendix ARadiotechnolgy
The purpose of a radio transmitter (fig. A.1)is to transform a signal to radiowaves,thus enabling a wireless signal transfer. This is done by utilizing the fact that analternating current creates a electromagnetic field around the antenna. This fieldemanates from the antenna, and can be received by another antenna.
Figure A.1: Function diagram of radio transmitter
To enable the transmission of multiple signals, the signals are modulated toanother frequency by either Amplitude Modulation (AM), Phase Modulation (PM)or Frequency Modulation (PM). Modulation tecniques are discussed in appendixC. This enables transmission of more than one signal at a time, as each signal has adifferent carrier frequency (fig. A.2). Due to the requirements of sound quality forcommercial radiostations, they require a bandwidth of 150kHz. The radio amateursonly require 16kHz, due to the fact that they do not need a high sound quality,as they only transmit speech. [Christensen, 1999] FiXme: find sidetal i HF-teknik,
eller en anden kilde
Figure A.2: Frequency overview of a FM-modulated signal.
The purpose of a radio receiver (fig. A.3) is to receive the transmitted signal,and transform it to its original form.
7
A.1. PURPOSE OF THE PROJEKT
Figure A.3: Function diagram of radio receiver
A.1 Purpose of the projekt
The purpose of this projekt is to convert a FM-modulated signal to sound. Thissignal has been chosen to have a carrier wave of 145MHz and a bandwidth of 16kHz.This is due to:
• The limited bandwidth of the soundcard (see appendix B for capabilities ofthe sound card, and appendix D.4 for the consequenses of downconversion)
• The 145MHz is useable by all for transmission, which will enable us to testthe system with a transmitted signal
8
APPENDIX B. SOUND CARD
Appendix BSound card
Sampling a signal using using a PC sound card is only possible if the sound cardsmeets certain specification set by the properties of the input signal. The inputsignal may also have to be adapted to meet the input specification of the soundcard. The purpose of this appendix is to examine the specification of a general PCsound card. Audio Codec ’97 is a royalty-free sound card standard developed byIntel Corporation. The specification defines the architecture and digital interfaceof a sound card including analog performance characteristics of the input signal.In order to sample the input signals there are a few key parameters the have tobe met by the AC’97 standard. These parameters include the bandwidth andthe sampling resolution. In the AC ’97 v2.3 Component Specification the keyparameters concerning the frequency response and sampling frequency is listed asin table B.1 Intel Corporation
Parameter Min typ Max UnitsSampling Frequency - - 48000 HzAnalog frequency responce ± 1 dB 20 - 20000 HzTransition band 19100 - 28800 HzStop band 28800 - - HzStop band rejection -74 - - dBGroup delay - - 1 ms
Table B.1: Frequency response of an AC’97 compliant sound card
Other parameters that have to be taken into consideration when sampling usingan AC’97 sound card is listed in table B.2 AC’97 defines the sampling resolution
Parameter Min typ Max UnitsMic full scale input voltage (20 dB boost) - 0.1 - VrmsMic full scale input voltage (0 dB boost) - 1 - VrmsInput imdance 10 - - kΩInput Capasitance - 7.5 - pF
Table B.2: Input parameters of an AC’97 compliant sound card
9
as full-duplex 16 bit.
10
APPENDIX C. MODULATION
Appendix CModulation
The purpose of a communication system is to transmit an information-bearingsignal from a transmitter to a receiver. The information-bearing signal is referredto as baseband signal, where the term baseband is used to designate the band offrequencies representing the original signal.
Often it is necessary to shift the baseband frequencies to a frequency range moresuitable for transmission, which is done by modulation. Modulation is a methodwhere a carrier frequency is changed in accordance to a modulation signal. Ascarrier frequency a sinusoidal wave is often used, in which case the modulationbecomes a continuous-wave modulation process. The modulation frequency is thebaseband signal, and the result of the modulation is referred to as the modulatedsignal.
In this report two types of modulation technics are described amplitude mod-ulation (AM) and frequency modulation (FM). In AM the modulation signal fm
modulates the amplitude of the carrier signal fc, and in FM the modulation sig-nal fm modulates the frequency of the carrier signal fc. In figure C.1 a AM- andFM-signal is shown. For the AM signal the frequency is determined by fc andthe amplitude is determined by fm, whereas for the FM signal the amplitude isdetermined by fc and the frequency is determined by fm. The appendix is basedon [Haykin, 2001, chapter ?] and [Johnson et al., 1999, Page 325 and chapter 17].
C.1 Amplitude modulation
AM is defined as a process in which the amplitude of the carrier wave fc is variedabout a mean value, linearly with the baseband signal fm. If the carrier- andbaseband-signal are given as
fc = Ac cos(ωct) (C.1)
fm = m(t) (C.2)
then the modulated signal can be described in its general form as
s(t) = Ac[1 + kam(t)] cos(ωct) (C.3)
11
C.1. AMPLITUDE MODULATION
0 2 4 6 8 10 12−1
0
1
t [s]A
mpl
itude
Carrier wave
0 2 4 6 8 10 12−1
0
1
t[s]
Am
plitu
deModulation signal
0 2 4 6 8 10 12−2
0
2
Am
plitu
de
t[s]
Amplitude modulated signal
0 2 4 6 8 10 12−1
0
1
Am
plitu
de
t[s]
Frequency modulated signal
Figure C.1: Illustration of the carrier wave, baseband signal, amplitude modulatedsignal and frequency modulated signal
0 2 4 6 8 10 12−2
0
2
Am
plitu
de
t[s]
Amplitude modulated signal Ka = 0.5
0 2 4 6 8 10 12−2
0
2
Am
plitu
de
t[s]
Amplitude modulated signal Ka = 1
0 2 4 6 8 10 12−5
0
5
Am
plitu
de
t[s]
Amplitude modulated signal Ka = 2
Figure C.2: Amplitude modulation of a signal using three different values of ka.Values above 1 result in over modulation and phase reversal
Where ka is a constant called the amplitude sensitivity. In equation C.3 it is clearthat the value of kam(t) determines the amplitude of the modulated signal. Thevalue of kam(t) must therefore be chosen suitable in order to avoid over modulationand phase reversal. Figure C.2 illustrate at AM signal with three different values
12
APPENDIX C. MODULATION
of ka and
m(t) = sin(1 · t) (C.4)
The upper graph illustrate the situation with ka = 0.5 where the original signalm(t) is clearly represented as a envelope of the AM signal. The middle graphillustrate the situation with ka = 1. As with ka = 0.5 the original signal is clearlyrepresented as a envelope, but the modulation signal now completely eliminate thecarrier signal during it’s minimum level. The bottom graph illustrate the situationwith ka = 2 and the baseband signal is no longer an envelope of the AM signal,hence it follows that to avoid over modulation and phase reversal
|kam(t)| < 1. (C.5)
An other criteria for the modulation to be successful is
fc >> W (C.6)
where W is the highest frequency component of the baseband signal m(t). If thiscriteria is not fulfilled it will not be possible to visualize an envelope. The criteriais illustrated in Figure C.3, where the graphs on the left illustrate the AM usinga carrier frequency 20 times as fast as the baseband signal, resulting in a clearenvelope of the baseband, as in the bottom left graph. To the right a carrierfrequency only twice as fast as the baseband signal is used, and the basebandsignal can hardly be recognized in the AM signal as in the bottom right graph.
0 2 4 6 8 10 12
−0.5
0
0.5
1
t [s]
Am
plitu
de
Carrier wave (20 ω/s)
0 2 4 6 8 10 12
−0.5
0
0.5
t[s]
Am
plitu
de
Modulation signal (1 ω/s)
0 2 4 6 8 10 12
−1
0
1
Am
plitu
de
t[s]
Amplitude modulated signal
0 2 4 6 8 10 12
−0.5
0
0.5
1
t [s]
Am
plitu
de
Carrier wave (2 ω/s)
0 2 4 6 8 10 12
−0.5
0
0.5
t[s]
Am
plitu
de
Modulation signal (1 ω/s)
0 2 4 6 8 10 12
−1
0
1
Am
plitu
de
t[s]
Amplitude modulated signal
Figure C.3: Amplitude modulation using two different carrier frequencies.
In order to determine the frequencies contained in the modulated signal equationC.3 is rewritten, and m(t) is replaced by a sinusoidal wave.
s(t) = Ac cos(ωct) + Acka cos(ωct) sin(ωmt) (C.7)
13
C.2. FREQUENCY MODULATION
From equation C.7 it can be seen that the modulated signal contains the carrierfrequency fc and the frequencies of the product
Acka cos(ωct) sin(ωmt). (C.8)
In order to determine the frequencies of the product, the product is rewritten byEuler.
Acka
j4
(
ejωct + e−jωct)
·(
ejωmt − e−jωmt)
(C.9)
Acka
j4
(
ejωct · ejωmt − ejωct · e−jωmt + e−jωct · ejωmt − e−jωct · e−jωmt)
(C.10)
Acka
j4
(
ej(ωc+ωm)t − ej(ωc−ωm)t + ej(−ωc+ωm)t − ej(−ωc−ωm)t)
(C.11)
Acka
2
(
sin((ωc + ωm)t) + sin((−ωc + ωm)t))
(C.12)
Equation C.12 shows that the frequencies contained in the product is (ωc + ωm)and (−ωc + ωm). Figure C.4 shows the frequency spectrum of a AM signal where
fc = cos(2 · π · 200 · t)
fm = sin(2 · π · 5 · t)
ka = 0.5
Figure (a) shows that the three frequencies exist as both positive and negativefrequencies. This means that the frequency (−ωc + ωm) also exist as a positivefrequency (ωc − ωm) and (ωc + ωm) exist as a negative frequency (−ωc − ωm). Inpractice negative frequencies does not exist, and only the positive frequencies is leftas shown in figure C.4 (b). Figure (b) shows that the side frequencies (ωc + ωm)and (ωc −ωm) is placed on each side of the carrier frequency with a dictance of fm
C.2 Frequency modulation
FM is defined as a process in which the frequency of the carrier wave fc is variedabout a mean value, as a function of the amplitude for the baseband signal fm. Ifthe modulation signal is a sinusoidal signal defined as
m(t) = Am cos(ωmt) (C.13)
then the instantaneous frequency is defined by
fi(t) = fc + kfAm cos(ωmt) (C.14)
fi(t) = fc + ∆f cos(ωmt) (C.15)
Where
∆f = kfAm (C.16)
14
APPENDIX C. MODULATION
185 190 195 200 205 2100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 (b)
frequency (Hz)
S(f
)
−400 −200 0 200 4000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
4 (a)
frequency (Hz)
S(f
)
Figure C.4: (a) Spectrum of the AM signal (b) Detailed spectrum of the positivefrequencies
The unit ∆f is the frequency deviation, representing the frequency variation fromthe carrier frequency. The characteristic for FM is that ∆f is proportional withthe amplitude of the modulation signal, and independent of the frequency of themodulation signal. To get a expression for how much the frequency changes over aperiode from 0 to t equation C.15 is integrated.
Θi(t) = 2π
∫ t
0
fi(τ)dτ (C.17)
Θi(t) = 2πfct +∆f
fm
sin(ωmt) (C.18)
The ration of the frequency deviation ∆f , to the modulation frequency is calledthe modulation index which is given as
β =∆f
fm
(C.19)
If the modulation index is substituted into equation C.18 a new expression for Θi(t)is achieved.
Θi(t) = ωct + β sin(ωmt) (C.20)
From equation C.20 it can be seen that the β represents the phase deviation of theFM signal. This means that β represents the maximum variation from the angleωct. The FM signal can now be expressed as
s(t) = Ac cos[
ωct + β sin(ωmt)]
(C.21)
Determining the spectrum of a FM signal is not as easy as it was for a AM signal.This is because an FM signal modulated by a sinusoidal wave shown in equationC.21 is not a periodic function unless the carrier signal frequency fc is a integral
15
C.2. FREQUENCY MODULATION
multiple of the modulation signal fm. A different method to determine the spectrumis therefore necessary. By rewriting equation C.21 the following expression for s(t)can be achieve
s(t) = Re[
Acejωct+jβ sin(ωmt)
]
(C.22)
s(t) = Re[
s(t)ejωct]
(C.23)
Where s(t) is a complex envelope of the FM signal s(t), given by
s(t) = Acejβ sin(ωmt) (C.24)
Unlike the FM signal, the complex envelope s(t) is periodic over time. This meansthat s(t) can be expanded by the complex fourier series, which is defined as
s(t) =
∞∑
n=−∞
cnejωmnt (C.25)
where the complex fourier coefficient cn is defined by
cn =1
T
∫ T2
−T2
s(t)e−jωmntdt (C.26)
Inserting the value of s(t) from equation C.24 into equation C.26
cn =Ac
2π
∫ π
−π
ejβ sin(ωnt) · e−jωntdt (C.27)
cn =Ac
2π
∫ π
−π
ejβ sin(ωnt)−jωntdt (C.28)
Defining a new variable x = ωmt and inserting it in equation C.28 gives
cn =Ac
2π
∫ π
−π
ej(β sin(x)−nx)dx (C.29)
Except for Ac equation C.29 can be recognized as the n‘th order Bessel function ofthe first kind and argument β, which is commonly denoted as
Jn(β) =1
2π
∫ π
−π
ej(β sin(x)−nx)dx (C.30)
This gives a new expression for cn
cn = Ac · Jn(β) (C.31)
Substituting equation C.31 in equation C.25
s(t) = Ac
∞∑
n=−∞
Jn(β)ejωmnt (C.32)
16
APPENDIX C. MODULATION
Now substituting the new expression for s(t) back into equation C.23 gives
s(t) = Ac · Re[
∞∑
n=−∞
Jn(β)ej(ωc+nωm)t]
(C.33)
By remembering that
Re[
ej(ωc+nωm)t]
= cos[
(ωc + nωm)t]
(C.34)
equation C.33 can be rewritten to
s(t) = Ac
∞∑
n=−∞
Jn(β) · cos[
(ωc + nωm)t]
(C.35)
Fourier transform of equation C.35 is done by using the cosine fouries transformpair, thus the frequency spectrum of a FM signal equals
S(t) =Ac
2
∞∑
n=−∞
Jn(β) ·[
δ(f − nfm − fc) + δ(f + nfm + fc)]
(C.36)
A plot of the Jn(β) is shown in Figure C.5. The first 4 orders are plotted with β
0 5 10 15−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
β
J nβ
0 order1st order2nd order3rd order
Figure C.5: Besselfunktion of 0 to 3 orders as a funtion of β
values from 0 to 10, however when modulation an base band signal a fixed β is used.Figure C.6 show the first 25 orders of the besselfunction with β = 1 5 15. Fromequation C.19 it is clear then when frequency modulation a signal tone base bandsignal the frequency deviation ∆f is increased when β is increased, as illustratedin the figure. As a result of equation C.16 the amplitude of the FM modulated
17
C.2. FREQUENCY MODULATION
−25 −20 −15 −10 −5 0 5 10 15 20 25−0.5
0
0.5
1
n
J n(1)
−25 −20 −15 −10 −5 0 5 10 15 20 25−0.4
−0.2
0
0.2
0.4
n
J n(5)
−25 −20 −15 −10 −5 0 5 10 15 20 25−0.4
−0.2
0
0.2
0.4
n
J n(15)
Figure C.6: First 25 orders of the besselfunction with 3 fixed β values
signal variate. The opposite is often the case; when frequency modulating, the baseband signal often contain multiple frequency components and the amplitude ofthe modulation signal is kept constant, thereby keeping ∆f constant. When ∆f isconstant and β is increased there is no longer fm between each frequency componentin the fm signal. This result in a reduced bandwidth of the fm signal needed tomodulate a given base band signal. As β approaches infinity the bandwidth of thefm signal approaches 2 · ∆f . This rule is knowned as Carson’s rule
BT = 2 · ∆f
(
1 +1
β
)
(C.37)
18
APPENDIX D. DOWNCONVERTER
Appendix DDownconverter
A modulated radio signal that is received through an antenna, sr(t), has two im-portant caracteristics; a carrier frequency fc and a signal bandwidth fBW. Thebandwidth is centered around the carrier frequency as illustrated in figure D.1. Inorder to restore the message signal from the modulated signal, it is conveinientto translate the signal down in frequency by some fdis as also shown in figureD.1. Displacement of a signal spectrum is referred to as frequency translation andtranslating a signal down in frequency is referred to as frequency-down conversion.[Haykin, 2001, page 103]
Figure D.1: Displacement of frequency spectrum.
D.1 Requirements
In this particular project the requirement to downconversion is to displace a smallspectrum FM signal so that the signal can be sampled using an AC97 compliantsound card, described in appendix B, which yields the specifications mentioned intable D.1. The received signal will be the one described in appendix A.1, hence thecarry frequency is approximately known and so is the bandwidth of the signal. Aquick calculation yields that it should be possible to fit the received signal into thevalid band of input frequencies of the sound card. In the design it is assumed thatthe received radio channel does not have any neighbourgh channels.
19
D.2. BUILDING BLOCKS
Parameter Value Units
Received signal fc 144 MHzReceived signal fBW 16 kHzDownconverted signal fmin 20 HzDownconverted signal fmax 19.2 kHz
Table D.1: Specifications of input and output of the downconverter.
D.2 Building blocks
Downconverters are very common and despite the fact that they can be constructedin many ways, they all consist of a few simple building blocks, which will be de-scribed in the following.
D.2.1 Mixer
The mixer is essentially a product modulator, that multiplies the received signalsr(t) with a local oscillator (LO) signal Ac cos(ωLOt). As a result of this, thespectrum of sr(t)is moved along the frequency axis with fLO. The amplitudeof the translated signal will be Acsr(t). Because the phase of the mixed signalreverses whenever the received signal sr(t) crosses zero, the new spectrum has amirror image around the frequency fLO. In practice this means that whenevera signal is mixed in order to translate it in frequency, two spectras are created- each with the same bandwith but with different carrier frequencies; the one isthe received signal shifted downwards and the other is the received signal shiftedupwards in frequency. This is explained in appendix FiXme: Ref til Jes’s afsnit
om sampling and also sketched in figure D.2 and shown mathemathically in sectionD.4. [Haykin, 2001, pages 94-95,103-104]
f1 = fc − fLO (D.1)
f2 = fc + fLO (D.2)
Figure D.2: Mirrors as a result of product modulation.
Refering to figure D.2, it is possible to chose fLO so that the spectrum near −f1
overlaps with the spectrum near f1. This is called sideband overlap and basically
20
APPENDIX D. DOWNCONVERTER
turns the signal into a mess. This is also why it is not trivial to build zero IF
receivers, i.e. converting directly to baseband. [Laskar et al., 2004, pages 32-44]
Mixers are available as premanufactured building blocks for various frequencyranges. A mixer has 3 terminals; RF (Radio Frequency), LO (Local Oscillator) andIF (Intermediate Frequency). The block diagram symbol of a mixer is shown infigure D.3.
Figure D.3: Symbol of a 3-terminal mixer.
D.2.1.1 Filter
To perform image rejection on either the down converted or up converted signal,a bandpass filter can be applied. The filter should have a midband frequency neareither f1 or f2 depending on which one is wanted, and a bandwidth equal to thebandwidth of the signal fBW . Most modern wireless standards require 60 - 90 dBof image rejection. [Laskar et al., 2004, page 30]
Filters are available as premanufactured building blocks, with various standardintermediate frequencies. There are other building blocks that can be utilized indownconverters, e.g. phase-locked loops (PLLs) and phase shifters. Since these arenot nescessary for this application, they will not be discussed further.
D.3 Block diagram
Often the design of the downconverter is closely related to the demodulator thattypically follows it, and they are often very integrated, as the downconverter doesa part of the demodulation or prepares the signal for a particular demodulation.
In this project the aim is to keep as much of the processing as possible inthe digital domain, and therefore it is desireable to capture the received signalas unprocessed as possible, hence simplifying the downconverter. One solution isshown in the block diagram of figure D.4.
Between the antenna and the first mixer some amplification will be nescessary(RF AMP), however the strength of the received signal is not known FiXme: at
this point .
The downconversion is done in two steps using two mixers. At the output ofeach mixer a filter is applied for mirror selectivity. To interface correctly with thesound card described in chapter B, the filter should contain an adjustable gain
21
D.4. MATHEMATICAL ANALYSIS
Figure D.4: Block diagram of a possible downconverter solution.
D.4 Mathematical analysis
The purpose of the downconverter is to translate the frequencies of the RF signal,in into baseband signals. Which can be sampled by an soundcard. The input signalto the downconverter will be in the form as shown in appendix C:
Ac · cos[
ωc · t + β · sin(ωc · t)]
(D.3)
At any give time the signal can be described as:
A1 · cos(ω1 · t) (D.4)
This signal is then multiplied with a cosine in a circuit know as a product mixerthis yields:
A1 · cos(ω1 · t) · A2 · cos(ω2 · t) (D.5)
By means of Euler this is:
A1 · A2 ·1
2· (ej·t·ω1 + e−j·t·ω1) ·
1
2· (ej·t·ω2 + e−j·t·ω2) ⇔(D.6)
A1 · A2
4·[
ej·t·ω1 · ej·t·ω2 + e−j·t·ω1 · ej·t·ω2 + ej·t·ω1 · e−j·t·ω2 + e−j·t·ω1 · e−j·t·ω2
]
⇔(D.7)
A1 · A2
2·[
cos[(ω1 + ω2) · t] +1
2·(
e−j·t·ω1 · ej·t·ω2 + ej·t·ω1 · e−j·t·ω2
)]
⇔(D.8)
A1 · A2
2·[
cos[(ω1 + ω2) · t] + cos[(ω1 − ω2) · t]]
(D.9)
The result of the mixing is two frequency components, the sum and differencefrequencies. By bandpass filtering at (ω1 −ω2) the incoming signal is translated toa lower frequency. After the filter only the difference signal will be left, as the filterwill attenuate the sum signal. Next the new signal is again mixed with a cosine atfrequency ω3:
A1 · A2
2· cos[(ω1 − ω2) · t] · A3 · cos(ω3 · t) (D.10)
The result of this mixing is:
A1 · A2 · A3
4·[
cos[(ω1 − ω2 − ω3) · t] + cos[(ω1 − ω2 + ω3) · t]]
(D.11)
22
APPENDIX D. DOWNCONVERTER
After a bandpass or lowpass filter only the low frequency components are passedyielding:
A1 · A2 · A3
4· cos
[
(ω1 − ω2 − ω3) · t]
(D.12)
Remembering that the input signal was substituted with the instant case thisyields:
Ac · A2 · A3
4· cos
[
(ωc − ω2 − ω3) · t + β · sin(ωm · t)]
(D.13)
D.5 Dimensioning
As the intermediate frequency between the mixers the frequency 70 MHz, is chosen.The LO frequencies is determined, selecting the center frequency of the downcon-verted signal to approximately the center frequency of the sound cards band ofvalid input frequencies, 9 kHz:
fLO,1 = 75 MHz (D.14)
fLO,2 = 145 · 106 − 75 · 106 − 9 · 93 = 69.991.000 Hz (D.15)
As generators for fLO,1 and fLO,2, laboratory RF signal generators are used.The model names of the ones used can be found in appendix F. The mixers arechosen regarding to the needed frequency ranges. The types used are mentioned intable D.2.
Manufacturer Model No. LO/RF IF UnitsMini-Circuits ZFM-3 0.04 - 400 DC - 400 MHzMini-Circuits ZFM-15 10 - 3000 10 - 800 MHz
Table D.2: Specifications of mixers.
As the filter following the first mixer the component mentioned in table D.3 isused. Since the attenuation in the stop band is not specified by the manufacturerit might not be sufficient.
Manufacturer Model No. Type fc fBW SectionsTexscan (Trilithic) 3BC 70/5-3-KK BP 70 MHz 5 MHz 3
Table D.3: Specifications of bandpass filter.
To filter the output from the second mixer a bandpassfilter is needed. As centerfrequency 9 kHz is chosen because it is the carrier frequency of the downconvertedsignal. The bandwidth should be 16 kHz and at the nyquist frequency, 24 kHz,the amplitude should be damped 40 dB. The design of the filter is described inappendix G.
23
D.6. SIMULATION
D.6 Simulation
The downconverter is simulated using Simulink. The reason for this is to examinethe waveforms at different points in the downconverter to improve the understand-ing. The model will not be accurate according to the downconverter that is built,because several parameters are left out and because the responses of the filters usedin the physical implementation are not known in detail. In the simulation only themixing process is taken into account, no gain considerations.
The downconverter can be modelled as shown in figure D.5. The mixers aremodelled using a product block. A Chebychew 1 bandpassfilter is selected, havingan order of 3, a passband between 424.115 · 106 rad/sec and 455.531 · 106 rad/secand a passband ripple of max. 0.5 dB. For the second filter a Bessel lowpass filteris selcted, having an order of 9 and a critical frequenzy of 376.99 · 106 rad/sec. Thesignal generators SG1, LO1 and LO2 all produce sine waves with amplitudes of 1and frequencies of 145 MHz, 75 MHz and 69.991 MHz respectively. The simulationparameters are shown in table D.4.
Tab Parameter Value
Solver Stop time 2e-4Solver Type Fixed-step Dormand-PrinceSolver Fixed step size 0.2e-9Workspace I/O Limit data points... UncheckWorkspace I/O Decimation 1Solver Stop time 0.0002
Table D.4: Simulation parameters that vary from default.
Figure D.5: Simulink model block diagram of downconverter.
The required sample frequency is calculated as
fs > 2 · (145 · 106 + 75 · 106) (D.16)
> 440 MHz (D.17)
24
APPENDIX D. DOWNCONVERTER
due to the fact that the highest frequency is the input frequency of 145 MHzshifted 75 MHz upwards. Converted to a maximum sampling time of 2.2727 · 10−9
s, the requirement is shown to be met by a factor of 10, comparing with table D.4.
The waveforms the simulation is shown in figure D.6. Figure (a) shows thewaveforms of the signal on both sides of the bandpass filter. FiXme: fejl i legend
+ forkert sprog The dotted line shows the mixed but unfiltered signal. The curvesare not smooth due to the sampling frequency, however this does not effect thecalculation. The solid line shows the filtered signal. If the filtered signal was shownin a longer time period it would represent a sine wave, which is the input signalshifted downwards. This signal is mixed again, and the results of this is shown infigure (b). Again the waveforms show that the filtering eliminate the high frequencycomponents and leave the downconverted signal, the solid waveform. In plot (c)the same waveform is shown in a longer time period, revealing a sine wave with afrequency of 9 kHz as expected.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−8
−0.5
0
0.5
1(a)
time (s)
S(t
)
Før BPEfter BP
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10−8
−0.4
−0.2
0
0.2
0.4(b)
time (s)
S(t
)
Før LPEfter LP
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−6
−0.4
−0.2
0
0.2
0.4(c)
time (s)
S(t
)
Efter LP
Figure D.6: Waveforms of simulation (a) after the first mixing, (b) after thesecond mixing and (c) the resulting output.
D.7 Test
The test described in appendix F showed that the downconverter is able to translateRF signals near 145 MHz to a low IF of 9 kHz. Furthermore the insertion loss ismeassured to 6.46 dB with an input level of -30 dBm. The insertion loss may varyfor other input levels.
25
Appendix ESampling
The goal of signal sampling, from the continious domain to the discrete domain, isto produce a data representation suitable for use in computers.
This chapter will introduce the concept of sampling, and its unwanted propertyof frequency aliasing. The chapter uses theory from [Oppenheim & Schafer, 1998,Section 4].
E.1 From continuous to discrete
The discrete data representation of the continuous signal is attained, by multiplyingthe input signal (xc(t)) with Diracs deltafunction (δ(t − nT )) at specific intervals.With n as all natural numbers and T = 1
fswith fs as the sampling frequency. The
process is shown in figure E.1.
Figure E.1: Principal sampling block. Showing input xc(t) multiplied with δ(t −nT ), producing the output x[n].
As δ(t − nT ) is 1 at t − nT = 0 and otherwise 0, the attained signal is:
x[n] = xc(t − nT ) (E.1)
Figure E.2 on page 29 shows the process in more steps.
E.2 Frequency attributes
Examining the preceeding process with the fourier transform, shows that “signalmirrors” are produced in the frequency domain. This occurs because a multipli-
26
APPENDIX E. SAMPLING
cation in time equals a convolution in frequency. This will be shown in the following.
The “sampling signal” (δ(t − nT )) can be represented as
s(t) =
∞∑
n=−∞
δ(t − nT ) (E.2)
with attributes
fs =1
T[Hz] Ωs =
2 · π
T
[
rad
s
]
Using the multiplication as shown in figure E.1 on the preceding page produces thefollowing
xs(t) = xc(t) · s(t) (E.3)
xs(t) =
∞∑
n=−∞
xc(t) · δ(t − nT ) (E.4)
xs(t) =
∞∑
n=−∞
xc(nT ) · δ(t − nT ) (E.5)
Using the fourier transfor on (E.3) produces
Xs(jΩ) = Xc(jΩ) ∗ S(jΩ) (E.6)
Using the convolution theorem
f(t) ∗ g(t) =
∫ t
0
f(τ)g(t − τ)dt (E.7)
and the fourier transform
F [f(t)] = F (jω) =
∫
∞
−∞
f(t) · e−jωtdt (E.8)
produces ()
Xs(jΩ) =1
T
∞∑
k=−∞
Xc(jΩ − k · Ωs) (E.9)
It is seen that Ω is a continuous variable, while Ωs determines the offset from theoriginal function because of the multitude of k’s. This is shown in figure E.3 onpage 30
If we examine figure E.4 on page 30 it is easily seen that if Ωs > 2 · Ωbw thesignals will overlap. This is called aliasing. To avoid aliasing, one must sample atat speed faster than twice the bandwidth of the input signal. This is called theNyquist frequency
Ωnyquist > 2 · Ωbw (E.10)
27
E.2. FREQUENCY ATTRIBUTES
Still missing from this chapter:
• Show nyquist, explain
• Add explanation of the step at () (and remove the cicle)
• Redraw graphs to proper scale
• first figure needs overhaul.. delta etc. . .
28
APPENDIX E. SAMPLING
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
Val
ue
Time
a)
Input signal
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
Val
ue
Time
b)
Input signalSampling signal
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Val
ue
Time
c)
Sampled signal
Figure E.2: a) The continous input signal b) Continous input signal multipliedwith sampling function c) Discrete time result
29
E.2. FREQUENCY ATTRIBUTES
Ωs Ωs
Xs(jΩ)
Ω
k = 1 k = 0 k = −1
Figure E.3: Frequency spectrum of sampled signal. The displacement by Ωs isshown for each k.
Xs(jΩ)
k = 1 k = 0
Ω
Ωs
Ωbw
Figure E.4: Superimposed frequency picture
30
APPENDIX F. MEASUREMENTS OF DOWNCONVERTER
Appendix FMeasurements of downconverter
The purpose of the measurements is to document that the downconverter is capableof translating radio frequency signals at fRF = 145 MHz down to fIF = 9 kHz. Thebandwidth of the signal is fBW = 16 kHz. The downconverter is being tested trans-lating a number of sinusoidal signals having fixed frequencies between fRF−0.5·fBW
and fIF + 0.5 · fBW. The frequency of the output signals is measured with an os-cilloscope, and it is verified that the waveforms have the correct frequency and form.
In the test the filter mentioned in table F.1 is used in stead of the analog filterdescribed in appendix G. The filters has an attenuation of less than 1 dB in thepass band and more than 40 dB in the stop band.
Manufacturer Model No. Type Passband Stop band Units
Mini-Circuits SLP-1.9 LP DC - 1.9 4.7 - 200 MHz
Table F.1: Specifications of lowpass filter.
F.1 Method
The test setup is shown i figure F.1. Note that the generators for the LO signalsare considered as part of system. The generator and indicators used are mentionedin table F.2.
Symbol Type Model Manufacturer AAU-nr
SG1 Signal Generator 2022 Marconi 08158IND1 Oscilloscope 2254A Tektronix 08388LO1 Signal Generator 2022D Marconi 33336LO2 Signal Generator 2022D Marconi 33337
Table F.2: Equipment used in test.
The test frequencies is chosen to cover the maximum and minimum frequenciesin a 16 kHz wide signal around 145 MHz:
31
F.2. RESULTS
Figure F.1: Test setup for downconverter.
f1 = fRF − 0.5 · fBW = 145 · 106 − 0.5 · 16 · 103 = 144.992.000 Hz (F.1)
f2 = fRF = 145 MHz (F.2)
f3 = fRF + 0.5 · fBW = 145 · 106 + 0.5 · 16 · 103 = 145.008.000 Hz (F.3)
The outcome of these three inputs should be 1 kHz, 9 kHz and 17 kHz.
1. Adjust the SG1 to a sine wave with with amplitude of -30 dBm, withoutmodulation.
2. Adjust the frequency of SG1 to f1.
3. On IND1, adjust the timebase and volt input attenuator to obtain the bestaccuracy.
Repeat above procedure for the three test cases.
F.2 Results
The results of the test is shown in table F.3.
Carrier frequency Output frequency Output level Time base Input attenuator
144.992 MHz 1.0 kHz 3.36 mV 200 µs 2 mV145.000 MHz 9.0 kHz 3.36 mV 20 µs 2 mV145.008 MHz 17.0 kHz 3.32 mV 10 µs 2 mV
Table F.3: Result of test.
The total voltage loss through the converter can be calculated as
VLoss = 20 · Log( Vin
Vout
)
[dB] (F.4)
The input power in Watts is given by
Pin = 10( inputlevel
10) · 1mW (F.5)
32
APPENDIX F. MEASUREMENTS OF DOWNCONVERTER
With an input level of -30 dBm this yields 1 µW, which is the equivalent of 7mV assuming 50 Ω impedance. Inserting the results of table F.3 and the 7 mV inequation (F.4) yields a loss of:
VLoss = 6.46 [dB] (F.6)
F.3 Conclusion
The test verifies that the downconverter is able to translate the RF signals to a lowIF of 9 kHz. The conversion loss of 6.46 might vary with different input levels dueto the non linear nature of the mixers. However it has not be possible to verifythis, as the oscilloscope is not able to measure voltages lower that the one used inthe test.
33
Appendix GFilter
bla bla bla
34
LIST OF CORRECTIONS
List of Corrections
FiXme: find sidetal i HF-teknik, eller en anden kilde . . . . . . . . . . . . 7FiXme: Ref til Jes’s afsnit om sampling . . . . . . . . . . . . . . . . . . . 20FiXme: at this point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21FiXme: fejl i legend + forkert sprog . . . . . . . . . . . . . . . . . . . . . . 25
35