Downconverter

A modulated radio signal that is received through an antenna, $s_\textrm{r}(t)$ , has two important characteristics; a carrier frequency $f_\textrm{c}$ and a signal bandwidth $f_{\textrm{BW}}$ . The bandwidth is centered around the carrier frequency as illustrated in figure C.1. In order to restore the message signal from the modulated signal, it is convenient to translate the signal down in frequency by some $f_{\textrm{dis}}$ as also shown in figure C.1. Displacement of a signal spectrum is referred to as frequency translation and translating a signal down in frequency is referred to as frequency down-conversion. [Haykin, 2001, page 103]

Figure C.1: Displacement of frequency spectrum.

Requirements

In this particular project the requirement for downconversion is to displace a small spectrum FM signal so that the signal can be sampled using an AC'97 compliant sound card, described in appendix A, which yields the specifications mentioned in table C.1. The received signal will be the one described in chapter 2, hence the carrier frequency is approximately known and so is the bandwidth of the signal. A quick calculation yields that it should be possible to fit the received signal bandwidth of 16 kHz, into the valid band of input frequencies of the sound card which is $\approx 19.2$ kHz. In the design it is assumed that the received radio channel does not have any neighbouring channels.

Table C.1: Specifications of input and output of the downconverter.

Parameter	Value	Units
Received signal $f_{\textrm{c}}$	145	MHz
Received signal $f_{\textrm{BW}}$	16	kHz
Downconverted signal $f_{\textrm{min}}$	20	Hz
Downconverted signal $f_{\textrm{max}}$	19.2	kHz

Building blocks

Downconverters are very common and despite the fact that they can be constructed in many ways, they all consist of a few simple building blocks, which will be described in the following.

Mixer

The mixer is essentially a product modulator, that multiplies the received signal $s_\textrm{r}(t)$ with a local oscillator (LO) signal $A_\textrm{c}\cos(\omega_{\textrm{LO}}t)$ . As a result of this, the spectrum of $s_\textrm{r}(t)$ is moved along the frequency axis by $f_{\textrm{LO}}$ . The amplitude of the translated signal will be $(A_\textrm{c}/2)s_\textrm{r}(t)$ . Whenever a signal is mixed in order to translate it in frequency, two spectras are created - each with the same bandwidth but with different carrier frequencies; the one is the received signal shifted downwards and the other is the received signal shifted upwards in frequency. This is sketched in figure C.2 and shown mathemathically in section C.4. [Haykin, 2001, pages 94-95,103-104]

$$f_1 = f_c - f_{LO}$$ (C.1)

$$f_2 = f_c + f_{LO}$$ (C.2)

Figure C.2: Mirrors as a result of product modulation.

Refering to figure C.2, $f_{LO}$ should be chosen so that the spectrum near $-f_1$ does not overlap with the spectrum near $f_1$ , because sideband overlapping complicates the situation significantly. This is also why it is not trivial to build zero IF receivers, i.e. conversion directly to baseband. [Laskar et al., 2004, pages 32-44]

Mixers are available as premanufactured building blocks for various frequency ranges. A mixer has 3 terminals; RF (Radio Frequency), LO (Local Oscillator) and IF (Intermediate Frequency). The block diagram symbol of a mixer is shown in figure C.3.

Figure C.3: Symbol of a 3-terminal mixer.

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 near either $f_1$ or $f_2$ depending on which one is wanted, and a bandwidth equal to the bandwidth of the signal $f_{BW}$ . Most modern wireless standards require 60 - 90 dB of image rejection. [Laskar et al., 2004, page 30]

Filters are available as premanufactured building blocks, with various standard intermediate frequencies. There are other building blocks that can be utilized in downconverters, e.g. phase-locked loops (PLLs) and phase shifters. Since these are not nescessary for this application, they will not be discussed further.

Block diagram

Often the design of the downconverter is closely related to the demodulator that typically follows it, and they are often very integrated, as the downconverter does a 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 in the digital domain, and therefore it is desireable to capture the received signal as unprocessed as possible, hence simplifying the downconverter. One solution is shown in the block diagram of figure C.4.

Figure C.4: Block diagram of a possible downconverter solution.

The downconversion is done in two steps using two mixers. At the output of each mixer a filter is applied for mirror selectivity. To interface correctly with the sound card described in appendix A, the filter should be followed by an automatic gain control (AGC).

Mathematical analysis

The purpose of the downconverter is to translate the RF bandpass signal, into a low IF bandpass signal, which can be sampled by an soundcard. The input signal to the downconverter will be in the form as shown in appendix B:

$$s(t)=A_{c} \cos \Bigl[\omega_{c} \cdot t + \beta \sin(\omega_{c} t) \Bigl]$$ (C.3)

At any give time the signal can be described as:

$$s_i(t)=A_{1} \cos (\omega_{1} t)$$ (C.4)

This signal is then multiplied with a cosine in a circuit known as a product mixer. This yields:

$$A_{1} \cdot \cos (\omega_{1} \cdot t) \cdot A_{2} \cdot \cos (\omega_{2} \cdot t)$$ (C.5)

By means of Euler this is:

$$\frac{1}{2} A_{1} A_{2} \cdot (e^{j \cdot t \cdot \omega_{1}} + e... ...1}{2} \cdot (e^{j \cdot t \cdot\omega_{2}} + e^{-j \cdot t \cdot \omega_{2}}) =$$ (C.6)

$$\frac{A_{1} A_{2}}{4} \Bigl[e^{j \cdot t \cdot \omega_{1}} e^{j \... ...{2}} + e^{-j \cdot t \cdot \omega_{1}} e^{-j \cdot t \cdot \omega_{2}} \Bigl] =$$ (C.7)

$$\frac{A_{1} \cdot A_{2}}{2} \cdot \Bigl[\cos[(\omega_{1}+\omega_{... ... \cdot t \cdot \omega_{1}} \cdot e^{-j \cdot t \cdot \omega_{2}}\Bigl) \Bigl] =$$ (C.8)

$$\frac{A_{1} \cdot A_{2}}{2} \cdot \Bigl[\cos[(\omega_{1}+\omega_{2}) \cdot t] + \cos[(\omega_{1}-\omega_{2}) \cdot t]\Bigl]$$ (C.9)

The result of the mixing is two frequency components, the sum and difference frequencies. By bandpass filtering at $(\omega_{1}-\omega_{2})$ the incoming signal is translated to a lower frequency. Next the new signal is again mixed with a cosine at frequency $\omega_{3}$ :

$$\frac{A_{1} \cdot A_{2}}{2} \cdot \cos[(\omega_{1}-\omega_{2}) \cdot t] \cdot A_{3} \cdot \cos(\omega_{3} \cdot t)$$ (C.10)

The result of this mixing is:

$$\frac{A_{1} \cdot A_{2} \cdot A_{3}}{4} \cdot \Bigl[\cos[(\omega_... ...-\omega_{3}) \cdot t] + \cos[(\omega_{1}-\omega_{2}+\omega_{3}) \cdot t] \Bigl]$$ (C.11)

After a bandpass or lowpass filter only the low frequency components are passed yielding:

$$\frac{A_{1} \cdot A_{2} \cdot A_{3}}{4} \cdot \cos \Bigl[(\omega_{1}-\omega_{2}-\omega_{3}) \cdot t \Bigl]$$ (C.12)

Remembering that the input signal was substituted for equation (C.4) this yields that the output signal will be:

$$\frac{A_{c} \cdot A_{2} \cdot A_{3}}{4} \cdot \cos \Bigl[(\omega_{c}-\omega_{2}-\omega_{3}) \cdot t + \beta \cdot \sin(\omega_{m} \cdot t) \Bigl]$$ (C.13)

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 downconverted signal to approximately the center frequency of the sound cards band of valid input frequencies, $9$ kHz:

$$f_{LO,1} = 75\textrm{ MHz}$$ (C.14)

$$f_{LO,2} = 145\cdot 10^6 - 75 \cdot 10^6 - 9 \cdot 9^3 = 69.991.000\textrm{ Hz}$$ (C.15)

As generators for $f_{LO,1}$ and $f_{LO,2}$ , laboratory RF signal generators are used. The model names of the ones used can be found in appendix C.6. The mixers are chosen on the basis of the needed frequency ranges. The types used are mentioned in table C.2.

Table C.2: Specifications of mixers.

Manufacturer	Model No.	LO/RF	IF	Units
Mini-Circuits	ZFM-3	0.04 - 400	DC - 400	MHz
Mini-Circuits	ZFM-15	10 - 3000	10 - 800	MHz

The component mentioned in table C.3 is the used bandpass filter.

Table C.3: Specifications of bandpass filter.

Manufacturer	Model No.	Type	$\mathbf{f_c}$	$\mathbf{f_{BW}}$	Sections
Texscan (Trilithic)	3BC 70/5-3-KK	BP	70 MHz	5 MHz	3

The lowpass filter used is mentioned in table C.4 The attenuation is less than 1 dB in the pass band and more than 40 dB in the stop band.

Table C.4: Specifications of lowpass filter.

Manufacturer	Model No.	Type	Passband	Stop band	Units
Mini-Circuits	SLP-1.9	LP	DC - 1.9	4.7 - 200	MHz

Measurements of downconverter

The purpose of the measurements is to document that the downconverter is capable of translating radio frequency signals at $f_\mathrm{RF} = 145$ MHz down to $f_{\mathrm{IF}} = 9$ kHz. The bandwidth of the signal is $f_{\mathrm{BW}} =$ 16 kHz. The downconverter is being tested translating a number of sinusoidal signals having fixed frequencies between $f_{\mathrm{RF}} - 0.5\cdot f_{\mathrm{BW}}$ and $f_{\mathrm{IF}} + 0.5\cdot f_{\mathrm{BW}}$ . The frequency of the output signals is measured with an oscilloscope, and it is verified that the waveforms have the correct frequency and form.

Method

The test setup is shown i figure C.5. Note that the generators for the LO signals are considered as part of system. The generator and indicators used are mentioned in table C.5.

Figure C.5: Test setup for downconverter.

Table C.5: Equipment used in test.

Symbol	Type	Model	Manufacturer	AAU-nr
SG1	Signal Generator	2022	Marconi	08158
IND1	Oscilloscope	2254A	Tektronix	08388
LO1	Signal Generator	2022D	Marconi	33336
LO2	Signal Generator	2022D	Marconi	33337

The test frequencies is chosen to cover the maximum and minimum frequencies in a 16 kHz wide signal around 145 MHz:

$$f_1 = f_{\mathrm{RF}} - 0.5\cdot f_{\mathrm{BW}} = 145\cdot 10^6 - 0.5\cdot 16\cdot 10^3 = 144.992.000 \mathrm{ Hz}$$ (C.16)

$$f_2 = f_{\mathrm{RF}} = 145 \mathrm{ MHz}$$ (C.17)

$$f_3 = f_{\mathrm{RF}} + 0.5\cdot f_{\mathrm{BW}} = 145\cdot 10^6 + 0.5\cdot 16\cdot 10^3 = 145.008.000 \mathrm{ Hz}$$ (C.18)

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, without modulation.
2. Adjust the frequency of SG1 to $f_1$ .
3. On IND1, adjust the timebase and volt input attenuator to obtain the best accuracy.

Repeat above procedure for the three test cases.

Results

The results of the test is shown in table C.6.

Table C.6: Result of test.

Carrier frequency	Output frequency	Output level	Time base	Input attenuator
144.992 MHz	1.0 kHz	3.36 mV	200 $\mu$ s	2 mV
145.000 MHz	9.0 kHz	3.36 mV	20 $\mu$ s	2 mV
145.008 MHz	17.0 kHz	3.32 mV	10 $\mu$ s	2 mV

The total voltage loss through the converter can be calculated as

$$V_{\mathrm{Loss}} = 20 \cdot log_{10}\Bigl(\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}\Bigl) \mathrm{ [dB]}$$ (C.19)

In radio communication a unit of measure called dBm is often used this is defined by the logarithm to the input signal relative to 1 mW into the working impedance, as follows

$$X = 10 \cdot log_{10}\Bigl(\frac{P_{\mathrm{in}}}{10^{-3}}\Bigl) \left[\mathrm{ dBm }\right]$$ (C.20)

where:
        

$X$ is the input level.	[dBm]
$P_{\mathrm{in}}$ is the input power.	[W]

Solving for the input power yields

$$P_{\mathrm{in}} = 10^{\left(\frac{X}{10}\right)} \cdot 10^{-3} \left[\mathrm{ W }\right]$$ (C.21)

With an input level of -30 dBm this yields 1 $\mu$ W, which is the equivalent of 7 mV assuming 50 $\Omega$ impedance. Inserting the results of table C.6 and the 7 mV in equation (C.19) yields a loss of:

$$V_{\mathrm{Loss}} = 6.46 \mathrm{ [dB]}$$ (C.22)

Summary

The test described in appendix showed that the downconverter is able to translate RF signals near 145 MHz to a low IF of 9 kHz. Furthermore the insertion loss is meassured to 6.46 dB with an input level of -30 dBm. The insertion loss may vary for other input levels.