Fourier Analysis Explained (video) -- Understanding Mixers

Over the years we'vE had a lot of posts about Joseph Fourier: 

https://soldersmoke.blogspot.com/search?q=Fourier

Recently I've found myself mentioning him while explaining how the diode ring mixer in our 

high-school direct conversion receiver project works. 

I think the video above does a good job in explaining how Fourier and his math explain how our mixers work.

My Kind of Math! The Wooden Fourier Transform Machine

fourier analiser from Gymnasiumnovum on Vimeo.

Mixer math with plywood and gears.

http://www.instructables.com/id/Plywood-Math-Machine/

http://hackaday.com/2016/04/11/fourier-machine-mimics-michelson-original-in-plywood/#more-199177

http://hackaday.com/2014/11/18/harmonic-analyzer-mechanical-fourier-computer/

Understanding Fourier Transforms

Lots of wisdom and insight here:

http://www.jezzamon.com/fourier/index.htm

Strongly recommended for those trying to understand mixers and harmonics. 

But why? Why Can't I Listen to DSB (or AM) on my Direct Conversion Receiver?

I've said this before:  I just seems so unfair.  We just should be able to listen to DSB signals with our beautifully simple homebrew Direct Conversion receivers. I mean, building a DSB transmitter is a natural follow-on to DC receiver construction.  And we are using AM shortwave broadcast stations (Radio Marti --I'm looking at you)  to test our DC receivers for AM breakthrough.  But when we tune these stations in, they sound, well, awful.  So unfair!  Why?   Unfortunately it has to do with laws.  Laws of physics and mathematics.  Blame Fourier, not me.  

Over the years there has been a lot of handwaving about this problem.  From Doug DeMaw, for example: 

In his "W1FB's Design Notebook," Doug wrote (p 171):  "It is important to be aware that two DSSC (DSB) transmitters and two DC receivers in a single communication channel are unsatisfactory.  Either one is suitable, however, when used with a station that is equipped for SSB transmissions or reception. The lack of compatibility between two DSSC (DSB) transmitters and two DC receivers results from the transmitter producing both USB and LSB energy while the DC receiver responds to or copies both sidebands at the same time."

That's correct, but for me, that explanation didn't really explain the situation.  I mean we listen to AM signals all the time.  They produce two sidebands, and our receivers respond to both sidebands, and the results are entirely satisfactory, right?  Why can't we do this with our Direct Conversion receivers?  I struggled with this question before: https://soldersmoke.blogspot.com/2015/07/peter-parker-reviews-dsb-kit-and.html  You can see in that post that I was not quite sure I had the answer completely correct. 

It took some discussion with a fellow Vienna Wireless Society member, and some Googling and Noodling for me to figure it out.  But I think I've got it: 

Imagine a station transmitting a DSB signal at 7100 kHz with a 1 kHz tone at the AF input.  There will be signals at 7101 kHz and at 7099 kHz.  Assume the carrier is completely suppressed. 

We come along with our DC RX and try to tune in the signal. 

Remember that they heart of the DC RX is a product detector, a mixer with the VFO (or PTO) running as close as we can get it to the suppressed carrier frequency (which we can't hear).  

Lets assume that we can somehow get our VFO or PTO exactly on 7100 kHz.  The incoming signals will mix with the VFO/PTO signal.  We are looking for audio, so we will focus on the difference results and ignore the sum results of the mixing.  

The difference between 7101 and 7000 is 1 kHz.  Great! And the difference between 7099 and 7000 is 1 kHz also.  Great again, right?  We are getting the desired 1 kHz signal out of our product detector, right?  So what's the problem?  

Here it is: SIDEBAND INVERSION.  Factoring in this part of the problem helps us see the cause of the distortion that plagues DSB-DC communication more clearly. 

Remember the Hallas Rule:   Whenever you subtract the modulated signal FROM the unmodulated signal, the sidebands invert.  So, in this case, we are subtracting that 7099 "lower sideband" signal FROM the 7100 VFO/PTO signal.  So it will invert.  It will become an upper sideband signal at 1 kHz.  We will have two identical 1 kHz signals at the output.  Perfect right?   Not so fast. Not so PERFECT really.  

The perfect outcome described above assumes that our VFO/PTO signal is EXACTLY on 7100 kHz.  And exactly in phase with the suppressed carrier of the transmitter.  But if it is even SLIGHTLY off, you will end up with two different output frequencies, signals that will move in and out of alignment, causing a wobbling kind of rapid fade-in, fade-out distortion.  You can HEAR this happening in this video by Peter Parker VK3YE, starting at 6:28:

And you can see it in this LTSpice simulation.  

This LTSpice model just shows two diode ring mixers.  The transmitter is on the top, the receiver is on the bottom. The transmitter has RF at 7100 kHz at L1 and audio at 1 kHz at R1.   The receiver has the VFO at 7100.001 L7,  DSB from the transmitter at L12 with audio appearing at R4.  It is instructive to watch the output as you move the VFO frequency.  If you move the VFO freq away from the transmit carrier osc frequency you will see the distortion.  Here is the netlist for the LTSpice simulation: 

http://soldersmoke.com/DSBDC.net

On paper, using simple mixer arithmetic, you can tell that it will be there. With the VFO/PTO just 1 Hz (that's ONE cycle per second) off, you will end up with outputs at 1.001 kHz and at .999 kHz.  Yuck.  That won't sound good. These two different frequencies will be moving in and out of alignment -- you will hear them kind of thumping against each other.    And that is with a mere deviation of 1 Hz in the VFO/PTO frequency!  We are scornful when the SDR guys claim to be able to detect us being "40 Hz off."  And before you start wondering if it would be possible to get EXACTLY on frequency and in phase, take a look at the frequency readout on my PTO.  

Now consider what would happen if the incoming signal were SSB, lets say just a tone at 7101 kHz.  We'd put our VFO at around 7100 kHz and we'd hear the signal just fine.  If we were off a bit we'd hear it a bit higher or lower in tone but there would be no second audio frequency coming in to cause distortion.  You can hear this in the VK3YE video:  When Peter switches to SINGLE Sideband receiver, the DSB signals sound fine. Because he is receiving only one of the sidebands. 

The same thing happens when we try to tune in an AM station using a Direct Conversion receiver:  Radio Marti sounds awful on my DC RX, but SSB stations sound great. 

My Drake 2-B allows another opportunity to explore the problem.  I can set the bandwidth at 3.6 kHz on the 2-B, and set the passband so that I will be getting BOTH the upper and the lower sidebands of an AM signal. With the Product Detector and the BFO on,  even with the carrier at zero beat  AM sounds terrible.  It sounds distorted.  But -- with the Product Detector and BFO still on --  if I set the 2-B's  passband to only allow ONE of the sidebands through,  I can zero beat the carrier by ear, and the audio sounds fine. 

There are solutions to this problem:  If you REALLY want to listen to DSB with a DC receiver, build yourself a synchronous detector that gets the your receivers VFO EXACTLY on frequency and in phase with the transmitter's oscillator.  But the synchronizing circuitry will be far more complex than the rest of the DC receiver. 

For AM, you could just use a different kind of detector.  That will be the subject of an upcoming blog post. 

Please let me know if you think I've gotten any of this wrong.  I'm not an expert -- I'm just a ham trying to understand the circuitry. 

Degenerative Feedback and Distortion Reduction

Continuing on negative feedback, on SolderSmoke I recently asked for help in understanding why negative feedback is said to "reduce distortion." Let me know if you think I'm on the right track. Thanks to all who sent e-mails.

Following Fourier’s advice, let’s think of distortion as an additional waveform riding along with our desired signal. In the diagram we have a 5X voltage amplifier with 20 mV at the input, let’s say that it produces a complex distorted waveform that consists of our desired 100 mV sine wave, along with an ugly 10 mV distortion signal.

The feedback network takes 10 percent of both signals and feeds them back to the input (with a 180 phase shift). At the input, for the desired signal, the 10 mV of feedback meets up with 30 mV of input signal (as in TM 11-455, I’ll keep outputs the same, but increase inputs); we end up with 20mV at the input to the amplifier device. This then goes through the 5X amp and we get our 100 mV output.

But look what happens to the ugly distortion signal: It arises IN the device. When the feedback portion of this distortion gets to the input, it does NOT meet up with an input signal. It just goes back through the amp. So the feedback network takes 10% of the 10 mV distortion, introduces a 180 phase shift and sends this 1 mV waveform through the 5X amp. At the output of the amp we can think of the original 10 mV of distortion combining with what is now a 5mV out of phase signal. In this case, half of the distortion signal is canceled. We can say that compared with the no-feedback amplifier, distortion has been reduced from 10% to 5%. We can say that this circuit discriminates against distortion signals that arise inside the device. The desired signal meets up with the input signal, cancels a portion of it, but then the remaining signal goes through the amp producing the desired amplified signal. But the distortion signal has nothing to meet at the input. It just goes through the amp and then cancels a portion of distortion signal at the output. More desired signal, less distortion.

WA8WDQ Builds OZ1JHM's Arduino CW Decoder (Video)

Bill, Pete:
I wanted to update you on my DC receiver progress.  While I'm still operationally proficient in CW, many of my friends are not.  So I thought it would be fun to add a CW decoder to my DC receiver. 

In my research for a solution, I ran across a sweet decoder I thought might be of interest to the SolderSmoke listeners.  OZ1JHM developed a totally software based decoder for Arduino that uses the Goertzel Algorithm.  This algorithm performs similarly to a Fast Fourier Transform but only for tone decoding at specific frequencies.  This limitation keeps the code small and fast making it perfect for microcontrollers like the Arduino. 

I was able to hack Hjalmar's code into mine and the result is CW decoder functionality in the receiver with no additional hardware!  But, the Arduino Uno's performance is limited so I need to dynamically switch between receiver VFO/control code and the CW decoder in order to preserve real-time performance.  This is only my first pass so perhaps I will find a way to optimize the code to more fully integrate the two.  I currently switch back and forth based on whether the VFO knob has been rotated or is idle.  This at least gives the illusion of real-time integration but makes it harder to tune in a signal for the decoder. 

Now that the Arduino Zero is available, I've been considering moving that direction to dramatically improve available horsepower.  This isn't the first time I've run out of gas with the Uno.  Now it's time to start working on a transmitter module for the radio :).  You know, even though I have an operational K3, I find myself reaching for this radio first.  Something magical about using something you've created :).  But hey, preaching to the choir!

Be sure to check out Hjalmar's site (http://www.skovholm.com   and 

http://skovholm.com/cwdecoder) for details on his design and a video demo.

Brad  WA8WDQ

Visualizing Harmonics and Square Waves

Very illuminating.  Very cool.   We want more!  Show us how a mixer works! 

From:
http://hackaday.com/2015/09/17/visualizing-the-fourier-transform/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+hackaday%2FLgoM+%28Hack+a+Day%29&utm_content=Netvibes


Our book: "SolderSmoke -- Global Adventures in Wireless Electronics" http://soldersmoke.com/book.htm Our coffee mugs, T-Shirts, bumper stickers: http://www.cafepress.com/SolderSmoke Our Book Store: http://astore.amazon.com/contracross-20

Building the Ceramic Discrete Direct Conversion Receiver #4 -- The Mixer

I think the most important stage of a direct conversion receiver is the mixer.   This is the stage that takes the RF energy coming in from the antenna and -- in one fell swoop -- turns it into audio.

It is important to understand how this happens.  I go into this in some detail in the SolderSmoke book.  To summarize: 

1) You have two signals going into a non-linear device.  The way in which the smaller signal passes through the device -- how much it is amplified or attenuated -- depends on the instantaneous value of the larger signal.  We are not just adding the two signals together.

2) The waveform that comes out will be a complicated repeating waveform.  We know from Fourier that any complicated repeating waveform can be broken down into sine wave components.

3) When you analyze the complicated repeating waveforms coming out of the mixer, you will find that the sine wave components include a frequency that is the sum of the two inputs and another that is the difference between the two.

So lets suppose we have a non-linear device.  We send in a signal from our oscillator at 7061 kHz. Coming in from the antenna we have a signal at 7060 kHz.   The non-linear device will produce outputs at 14121 kHz (sum)  and at 1 kHz (difference).  We are interested in the difference frequency.  We can HEAR that one.  We feed it into our audio amplifiers and we can copy the Morse Code coming in.  It will sound like a 1 kHz tone going on and off as the operator at the distant station presses his code key.  (We don't really have to worry about the 14121 kHz signal -- it is easily eliminated by filters and would never make it through our audio amplifiers.  And in any case we could not hear it.)

What can we use as a non-linear device?  In this receiver we will use diodes.  Diodes are  extremely non-linear devices. They can be used as on-off switches, with one of the signals determining if they are on (conducting) or off (not conducting).  When used like this they are "switching mixers." In essence, a larger,  controlling signal from the VFO will be turning the diodes on and off. Thus the signal coming in from the antenna will be chopped up by the switching action of the diode being turned on and off.  This is non-linear mixing at its most extreme.  It will definitely produce the sum and difference products we are looking for.

We could build the mixer with just one diode. You could apply the VFO signal to the diode to turn it on and off, and then feed the signal from the antenna into the same diode.   You would get the sum and the difference product out the other end.   You will see very simple direct conversion receivers intended for use in software defined radio schemes using just one diode. But this kind of circuit has a couple of serious shortcomingsq: it is susceptible to "AM breakthrough" and it is "lossy."

The circuit we are using addresses these problems by using two diodes.  To reduce loss, one conducts during half of the oscillator signal's cycle, the other during the other half.  Here LTSpice is ueful. You can model this mixer and see in the simulator how each of the diodes handles half of the oscillator RF cycle, with both contributing to the AF signal we want at the output (the difference frequency).   (The schematic above is from LTSpice but it is not ready for simulation.  For this you should replace the variable resistor with two fixed 500 ohm resistors, and add two oscillators -- one with the weak incoming RF signal and the other the strong local oscillator signal.)

The AM breakthrough problem is also addressed by the use of two diodes.  Here's the problem:  If you are on 40 meters, there will be strong shortwave AM broadcast signals coming in from your antenna.  Some will be so strong that they will get past your front-end filtering.  If you were using just one diode, that diode might demodulate the AM signal -- the AM carrier would mix with the AM sidebands and you would have an undesired audio signal heading for your AF amplifiers. Many of us have experienced this -- you are trying to listen to ham radio SSB signals, but you can hear China Radio International playing in the background. 

The two diodes take care of this easily. Look at the way an AM signal would reach the diodes. The carrier (and its sidebands) going through the top diode will be 180 degrees our of phase with the signal going into the lower diode. But the output of the diodes are joined together.  They will cancel out.  We say that for the RF signal coming through from the antenna, the circuit is "balanced."  That signal -- in this case the undesired AM signal -- will cancel out at the junction of the two diodes.

But to understand this circuit you must see what is NOT cancelled out.  The signal from the VFO is hitting each diode with the SAME polarity at the same time.  Look at the 1k variable resistor. So the signal from the VFO will NOT be cancelled out at the output.  Nor will the mixing products produced in the diodes.  That last sentence is the key to all of this.  The sum and difference products that result from the mixing of the signal from the antenna and the signal from the VFO SURVIVE.  They are not cancelled out.

We can easily select the one we want.  An RF bypass capacitor connected from the output of the mixer to ground will get rid of most of the VFO signal (7061 kHz) and most of the sum product (14121 kHz) while passing the audio to the AF amplifiers. 

When I built this detector I used a trifilar toroid out of a box of them that Farhan left with me back in May. I used two of the windings  secondary and one of the windings for the primary.  You might want to make a more simple transformer using an FT-43 type core.  I recommend W8DIZ as a source. 

I hope this explanation helps, and I hope I got it right.  Let me know if you see any errors in my explanation.  Tinker with the circuit when you build it.  You should be able to get it going.       

Complete Schematic

A Good Old VFO (by Rick, KK7B)

Here is another really great message from Rick, KK7B, sent to the emrfd yahoo group: [emrfd] A Good Old VFO Saturday, August 22, 2009 10:29 PM From: Rick To: emrfd@yahoogroups.com For several critical receiver applications in my lab I've used old Collins PTOs converted to solid state (I just replace the triode in the classic Hartley circuit with a J310 and run the circuit from a 9 volt regulator). I have half a dozen of them in dedicated propagation study receivers, and one SSB exciter I occasionally use on UHF. The other day I was changing something else in one of my receivers and connected the solid-state PTO to the frequency counter on my bench. The PTO was set to 3.100000 MHz. From a cold start (it hadn't been turned on for years) it drifted three Hz over the first ten minutes, and then a total of 10 Hz over the next few hours. When I calibrated one of my 144 MHz propagation study receivers 25 years ago, total frequency drift was <18Hz/hour. I expect most of that was aging of the overtone crystal oscillator in the premix circuit. Old Collins PTOs are common (someone at Dayton this year had a box of unknown ones in decent shape for $10 each, and there are R390 PTOs in the current Fair Radio flyer). I've never had one fail, tuning resolution is infinite, phase noise is low, digital noise is zero, and once I build one into a receiver, that part of the project is done--no improvements, software upgrades, needed. My research receivers are connected to a baseband Fourier analyzer (yes...even 25 years ago). The finest resolution I've used for serious experiments is 10 milliHertz, but more often I use 1 Hz resolution, with 1024 channels in the output spectrum. I often average spectra for more than a minute, so frequency drift needs to be less than 1 Hz per minute. The solid-state Collins PTO is much more stable than needed even for those critical experiments. This is not a fluke. Every Collins PTO I've converted to solid state using a U310 or J310 has had similar performance. Sometimes it is useful to remember that the major benefit of digital frequency synthesis is that it is quick, cheap, and frequency agile. No commercial manufacturer could afford to build a transceiver with a Collins Mil-Spec PTO in it these days. But for an amateur with mechanical skills or access to surplus hardware who needs just one good oscillator, the venerable Hartley with a temperature compensated tuned circuit and a JFET can provide outstanding performance. In music, art, architecture, automobiles, motorcycles. .. there are recognized "golden eras" where some combination of factors resulted in technical hardware that is widely recognized as being superior to what is being produced today. Often the difference is directly related to the amount of skilled labor needed during production. As technical hobbyists, we automatically assume that new is better, but as experimenters, we should be open to the idea that sometimes the technology, ideas, and block diagrams of an earlier era are superior to the cost-driven disposable technology coming off fully automatic assembly lines in some out-of-the-way place with inexpensive labor and attractive business tax codes. The idea that old technology designed decades ago by retired guys might be better than new technology is a radical concept in electronics. But NASA is using a brand new, hand built, Traveling Wave vacuum tube in the current Moon exploration mission. After 100 years of radio experiments- -it is fun to look back and find old technology that might actually work better than some of the new things we've been inventing recently. Best Regards, Rick KK7B

Are Diode Ring Mixers Fundamentally Different?

Joop, PE1CQP, and I have been discussing mixer circuits, especially the ever-popular diode ring.
Here is my latest e-mail to Joop. The RSGB diagram for the ring diode mixer appears above.

Joop: I think the way the diode ring mixer works is very different from the way a two diode singly balanced mixer functions. The effect, of course, is the same. But the polarity reversing element introduced by the ring configuration -- it seems to me -- makes this a very different circuit.

Attached is the RSGB Handbook diagram I mentioned. I like it, because you can really SEE how the actions of the diode ring produce the sum and difference freqs (you have to keep Fourier in mind, and imagine the results of filtering).

The two diode circuit simply "chops" the input signal at the rate of the LO. And it would even work in a non-switching mode -- you could, for example, use FETs instead of the diodes and bias them to operate in the non-linear portion of their curves, right? This makes me think that the diode ring mixer circuits (aka "polarity switching mixers" or "commutating mixers") are very different.

73 Bill

The Art of Electronics #5 Paul Horowitz on SETI (and lots of other radio stuff)

In 2016 Paul Horowitz  talked about SETI at Google. Fascinating stuff.  Paul did an especially good job of weaving in a lot of radio/electronic and computer info.  

-- I was pleased to learn that one of the early radio astronomy antennas used plywood covered with copper.  I hope it was copper tape! 

-- I didn't know that the Fast Fourier Transform was something developed in the 1960s. 

-- Parkes Telescope!  Yea! 

-- Paul's "chirping" of receivers to screen out targets that are NOT doppler shifting (i.e. terrestrial signals). 

-- Paul tells the group that "amateur" does not mean unprofessional -- it means that the person is doing it for the love of doing it.  Amen. 

-- SETI at Home. 

-- Tube op-amps!  (was that two 12AX7s?)  

--  A variometer!  Wow!  I have two here -- one in the ET-2 regen receiver  (a gift from Pericles HI8P), and another that I homebrewed using a 35mm film can.  

Great stuff from Paul.