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Superheterodyne mixer question
I'm trying to settle a debate with a friend, and my knowledge of
mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
Your friend is right.
If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
Your friend is right.
If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
Yes, your friend is right, but there is a grain of truth in what you are
saying also. A mixer multiplies two signals as your friend says and as said in the posting by W7EL. Signals can be represented by cosines, and the product of two cosines is: cosA * cosB = 0.5(cos(A-B) - cos(A+B) ) See product identities on http://www.swt.edu/slac/math/trigrev/trigrev.html and let A=2*pi*f1 and B=2*pi*f2. So while the operation of the mixer is that signals are _multiplied_, the frequencies will _add_ or subtract. -- Sverre Holm, LA3ZA --------------------------------- www.qsl.net/la3za |
Yes, your friend is right, but there is a grain of truth in what you are
saying also. A mixer multiplies two signals as your friend says and as said in the posting by W7EL. Signals can be represented by cosines, and the product of two cosines is: cosA * cosB = 0.5(cos(A-B) - cos(A+B) ) See product identities on http://www.swt.edu/slac/math/trigrev/trigrev.html and let A=2*pi*f1 and B=2*pi*f2. So while the operation of the mixer is that signals are _multiplied_, the frequencies will _add_ or subtract. -- Sverre Holm, LA3ZA --------------------------------- www.qsl.net/la3za |
Roy Lewallen wrote:
Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
Roy Lewallen wrote:
Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
Mixers are amplitude modulators.
One signal modulates the amplitude of the other. Trigonometrically we have - 2*Sin(A)*Sin(B) = Cos(A-B) - Cos(A+B) Where the A-B and A+B terms are appropriately described as "the products". In practice there are a great number of unwanted products output from a mixer because many harmonics of A and B are generated in the process and all continue to inter-modulate each other. The wanted product, the IF, is usually A-B or A+B. --- .................................................. .......... Regards from Reg, G4FGQ For Free Radio Design Software go to http://www.btinternet.com/~g4fgq.regp .................................................. .......... |
Mixers are amplitude modulators.
One signal modulates the amplitude of the other. Trigonometrically we have - 2*Sin(A)*Sin(B) = Cos(A-B) - Cos(A+B) Where the A-B and A+B terms are appropriately described as "the products". In practice there are a great number of unwanted products output from a mixer because many harmonics of A and B are generated in the process and all continue to inter-modulate each other. The wanted product, the IF, is usually A-B or A+B. --- .................................................. .......... Regards from Reg, G4FGQ For Free Radio Design Software go to http://www.btinternet.com/~g4fgq.regp .................................................. .......... |
"Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. Fred. |
"Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. Fred. |
Fred,
You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. As to Ian's comment...I don't think "adding" is the correct term either. "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. Joe W3JDR "Fred Bartoli" r_AndThisToo wrote in message ... "Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. Fred. |
Fred,
You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. As to Ian's comment...I don't think "adding" is the correct term either. "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. Joe W3JDR "Fred Bartoli" r_AndThisToo wrote in message ... "Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. Fred. |
"W3JDR" a écrit dans le message news: ... Fred, You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. Sure. Just try to suppress ear non linear effects and see how miserable composers will feel without it and how poor the music will sound to our "marvellous new ears". As to Ian's comment...I don't think "adding" is the correct term either. "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. Or, convolving, if the frequency domain, which tells all the story. Fred. |
"W3JDR" a écrit dans le message news: ... Fred, You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. Sure. Just try to suppress ear non linear effects and see how miserable composers will feel without it and how poor the music will sound to our "marvellous new ears". As to Ian's comment...I don't think "adding" is the correct term either. "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. Or, convolving, if the frequency domain, which tells all the story. Fred. |
Fred Bartoli wrote:
"Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. What the audio engineers do at the "mixing desk" involves only adding. What our ears do, is something else. But in fact, our ears are very close to linear. There is a belief that because we can hear "beat" frequencies, there must be some non-linear mixing in our ears... but that is actually a fallacy. The way we hear beat frequencies - the difference frequency between two separate audio tones - is due to simple linear addition and subtraction of two sound pressure waves. Non-linear mixing is not required. (If non-linear mixing were involved, we'd hear the sum frequency as well as the difference frequency... but in fact we don't, unless there is some other source of non-linearity outside of our ears.) -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
Fred Bartoli wrote:
"Ian White, G3SEK" a écrit dans le message news: ... Roy Lewallen wrote: Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! Well, ear is also somewhat non linear. So they are also doing mixing. What the audio engineers do at the "mixing desk" involves only adding. What our ears do, is something else. But in fact, our ears are very close to linear. There is a belief that because we can hear "beat" frequencies, there must be some non-linear mixing in our ears... but that is actually a fallacy. The way we hear beat frequencies - the difference frequency between two separate audio tones - is due to simple linear addition and subtraction of two sound pressure waves. Non-linear mixing is not required. (If non-linear mixing were involved, we'd hear the sum frequency as well as the difference frequency... but in fact we don't, unless there is some other source of non-linearity outside of our ears.) -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
W3JDR wrote:
Fred, You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. Hearing beats does not require non-linear or multiplicative mixing - please see my separate reply to Fred. As to Ian's comment...I don't think "adding" is the correct term either. I was referring to what *audio* engineers call "mixing", which is nothing else but simple adding or linear combining. I agree with everything you say below... "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. ...but the processes you describe are not what a straightforward audio "mixing" desk does. The device you describe above, an audio engineer would know as a "modulator" or a "ring modulator". For example, the LO could be at a low frequency, to get some kind of throbbing effect. Both RF and audio engineers would agree, that is true modulation. The difference is that RF engineers would also call that process "mixing"... but audio engineers would not because, to in their professional world, "mixing" means adding or linear combining. -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
W3JDR wrote:
Fred, You're exactly correct! That's why a piano tuner person can strike a tuning fork and a piano key at the same time and hear the frequency difference as a low beat note. Hearing beats does not require non-linear or multiplicative mixing - please see my separate reply to Fred. As to Ian's comment...I don't think "adding" is the correct term either. I was referring to what *audio* engineers call "mixing", which is nothing else but simple adding or linear combining. I agree with everything you say below... "Multiplying" or "sampling" are more precise terms. A perfect balanced unity-gain mixer actually uses one of the input signals to sample the other. On the positive half cycle of the LO, one phase of the RF signal is sampled, and on the other half cycle of the LO the opposite phase of the RF is sampled. Mathematically, this is equivalent to multiplying the RF signal by +1 or -1 on alternating half cycles of the LO. ...but the processes you describe are not what a straightforward audio "mixing" desk does. The device you describe above, an audio engineer would know as a "modulator" or a "ring modulator". For example, the LO could be at a low frequency, to get some kind of throbbing effect. Both RF and audio engineers would agree, that is true modulation. The difference is that RF engineers would also call that process "mixing"... but audio engineers would not because, to in their professional world, "mixing" means adding or linear combining. -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but: To be absolutely, mathematically correct, if you hold your mouth right, a "perfect" mixer with it's driving oscillator, is a linear device. It is _not_ a time-invariant device. It's linear because the IF signal that results from putting in the sum of any two RF signals is exactly equal to the sum of the IF signals that each result from each of the RF signals. If it were nonlinear then this would not be the case (and it wouldn't be a useful device for mixing). What gives a mixer it's "mixerness" is that it is linear but time-varying (output = input * some function of time). It is very easy to confuse time-varying linear with non-linear, and even easier in practice because in order to get the effect you need to use componant non-linearities to get the job done, just as you do with a class A amplifier. But it's usually harder to get the nonlinearities out of a mixer than an amplifier, so in real design you have to pay attention to non-linear effects like blocking and intermodulation in a mixer to a much greater extent than you do with an amplifier, and this reinforces the idea that a mixer is fundamentally nonlinear. This means that when you're analyzing a mixer (and ignoring real-mixer things like intermodulation) you can still use all the linear circuit theory stuff as long as you stay away from anything that depends on time-invariance. This means that _simple_ Laplace and Fourier analysis is out, but you can still use _careful_ Fourier analysis to figure out what the output will be for a given input and oscillator frequency. In fact, that's exactly what you are doing when you analyze a mixer: all of the desired behavior of a mixer can be exactly predicted with Fourier analysis. "Roy Lewallen" wrote in message ... Your friend is right. If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
Disclaimer (disflamer?): Everything that Roy says is true enough to get you
down the road of radio circuit design -- but: To be absolutely, mathematically correct, if you hold your mouth right, a "perfect" mixer with it's driving oscillator, is a linear device. It is _not_ a time-invariant device. It's linear because the IF signal that results from putting in the sum of any two RF signals is exactly equal to the sum of the IF signals that each result from each of the RF signals. If it were nonlinear then this would not be the case (and it wouldn't be a useful device for mixing). What gives a mixer it's "mixerness" is that it is linear but time-varying (output = input * some function of time). It is very easy to confuse time-varying linear with non-linear, and even easier in practice because in order to get the effect you need to use componant non-linearities to get the job done, just as you do with a class A amplifier. But it's usually harder to get the nonlinearities out of a mixer than an amplifier, so in real design you have to pay attention to non-linear effects like blocking and intermodulation in a mixer to a much greater extent than you do with an amplifier, and this reinforces the idea that a mixer is fundamentally nonlinear. This means that when you're analyzing a mixer (and ignoring real-mixer things like intermodulation) you can still use all the linear circuit theory stuff as long as you stay away from anything that depends on time-invariance. This means that _simple_ Laplace and Fourier analysis is out, but you can still use _careful_ Fourier analysis to figure out what the output will be for a given input and oscillator frequency. In fact, that's exactly what you are doing when you analyze a mixer: all of the desired behavior of a mixer can be exactly predicted with Fourier analysis. "Roy Lewallen" wrote in message ... Your friend is right. If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
Bill,
You said: " Any good technician will tell you it's an add and subtract process. Any good engineer will bore you to tears with complicated mathematical analysis. Guess which answer is more useful for your purpose?" Well of course...if you're only interested in what some of what comes out, then it's an 'add and subtract process'. But isn't that our initial definition of what we want a mixer to do? This is circular logic. You're chasing your own tail. The original question was more in the vein of 'by what mechanism does a mixer produce sum and difference frequency components'. The correct answer is that it implements the mathematical product of the two input signals, and that product contains sum and difference frequencies in addition to a host of other frequencies that includes the original frequencies, all their harmonics, and every conceivable product of those frequencies and their harmonics. It's not just a simple 'add and subtract'. It just so happens that we're most interested in the sum and difference, but there is much, much more going on. The "answer that is most useful for the purpose" is not necessarily the most simplistic. Consider the following profound statement from W.E. Deming: "If you can't describe what you are doing as a process, then you don't know what you are doing" Joe W3JDR |
Bill,
You said: " Any good technician will tell you it's an add and subtract process. Any good engineer will bore you to tears with complicated mathematical analysis. Guess which answer is more useful for your purpose?" Well of course...if you're only interested in what some of what comes out, then it's an 'add and subtract process'. But isn't that our initial definition of what we want a mixer to do? This is circular logic. You're chasing your own tail. The original question was more in the vein of 'by what mechanism does a mixer produce sum and difference frequency components'. The correct answer is that it implements the mathematical product of the two input signals, and that product contains sum and difference frequencies in addition to a host of other frequencies that includes the original frequencies, all their harmonics, and every conceivable product of those frequencies and their harmonics. It's not just a simple 'add and subtract'. It just so happens that we're most interested in the sum and difference, but there is much, much more going on. The "answer that is most useful for the purpose" is not necessarily the most simplistic. Consider the following profound statement from W.E. Deming: "If you can't describe what you are doing as a process, then you don't know what you are doing" Joe W3JDR |
You're absolutely correct. Production of new frequency components can be
done with either nonlinear or time-variant circuits. A square-law diode detector is an example of the first; a multiplier is an example of the second. I stand corrected -- thanks for pointing it out. Roy Lewallen, W7EL Tim Wescott wrote: Disclaimer (disflamer?): Everything that Roy says is true enough to get you down the road of radio circuit design -- but: To be absolutely, mathematically correct, if you hold your mouth right, a "perfect" mixer with it's driving oscillator, is a linear device. It is _not_ a time-invariant device. It's linear because the IF signal that results from putting in the sum of any two RF signals is exactly equal to the sum of the IF signals that each result from each of the RF signals. If it were nonlinear then this would not be the case (and it wouldn't be a useful device for mixing). What gives a mixer it's "mixerness" is that it is linear but time-varying (output = input * some function of time). It is very easy to confuse time-varying linear with non-linear, and even easier in practice because in order to get the effect you need to use componant non-linearities to get the job done, just as you do with a class A amplifier. But it's usually harder to get the nonlinearities out of a mixer than an amplifier, so in real design you have to pay attention to non-linear effects like blocking and intermodulation in a mixer to a much greater extent than you do with an amplifier, and this reinforces the idea that a mixer is fundamentally nonlinear. This means that when you're analyzing a mixer (and ignoring real-mixer things like intermodulation) you can still use all the linear circuit theory stuff as long as you stay away from anything that depends on time-invariance. This means that _simple_ Laplace and Fourier analysis is out, but you can still use _careful_ Fourier analysis to figure out what the output will be for a given input and oscillator frequency. In fact, that's exactly what you are doing when you analyze a mixer: all of the desired behavior of a mixer can be exactly predicted with Fourier analysis. "Roy Lewallen" wrote in message ... Your friend is right. If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
You're absolutely correct. Production of new frequency components can be
done with either nonlinear or time-variant circuits. A square-law diode detector is an example of the first; a multiplier is an example of the second. I stand corrected -- thanks for pointing it out. Roy Lewallen, W7EL Tim Wescott wrote: Disclaimer (disflamer?): Everything that Roy says is true enough to get you down the road of radio circuit design -- but: To be absolutely, mathematically correct, if you hold your mouth right, a "perfect" mixer with it's driving oscillator, is a linear device. It is _not_ a time-invariant device. It's linear because the IF signal that results from putting in the sum of any two RF signals is exactly equal to the sum of the IF signals that each result from each of the RF signals. If it were nonlinear then this would not be the case (and it wouldn't be a useful device for mixing). What gives a mixer it's "mixerness" is that it is linear but time-varying (output = input * some function of time). It is very easy to confuse time-varying linear with non-linear, and even easier in practice because in order to get the effect you need to use componant non-linearities to get the job done, just as you do with a class A amplifier. But it's usually harder to get the nonlinearities out of a mixer than an amplifier, so in real design you have to pay attention to non-linear effects like blocking and intermodulation in a mixer to a much greater extent than you do with an amplifier, and this reinforces the idea that a mixer is fundamentally nonlinear. This means that when you're analyzing a mixer (and ignoring real-mixer things like intermodulation) you can still use all the linear circuit theory stuff as long as you stay away from anything that depends on time-invariance. This means that _simple_ Laplace and Fourier analysis is out, but you can still use _careful_ Fourier analysis to figure out what the output will be for a given input and oscillator frequency. In fact, that's exactly what you are doing when you analyze a mixer: all of the desired behavior of a mixer can be exactly predicted with Fourier analysis. "Roy Lewallen" wrote in message ... Your friend is right. If you simply add or subtract two waveforms, no new frequencies are created. You end up with only the frequencies you started with and no more. (Theoretically, you could make one or more disappear if one of the added waveforms contained a precise negative of one or more frequency components of the other -- but you can never get any new frequencies.) That's because addition is a linear process, with linear having a precise definition that's appeared here a number of times before. (Subtraction is just addition, with one waveform inverted before adding.) Multiplication, though, is a nonlinear process by the precise definition used in circuit analysis, and it does create additional frequencies. Multiplying the two original signals of 1500 and 1955 generates the two new frequencies of 455 and 3455, for a total of four frequencies after multiplication. Adding them wouldn't do it. Most good mixers are actually more like switches than multipliers, but they're still nonlinear -- very much so -- and don't do anything remotely like adding the two signals. A doubly balanced mixer produces the sum and difference frequencies while not letting the original two frequencies get through to the output. The generation of the new frequencies by multiplication of the two originals is easily shown mathematically, as your friend says, with a short derivation by means of a trig identity. I'll be glad to post the derivation if you or other readers are interested, although it's widely available elsewhere. Roy Lewallen, W7EL Joer wrote: I'm trying to settle a debate with a friend, and my knowledge of mixers is pretty rusty. Say you have a receiver whose IF is 455 kHz, and it's tuned to a station at 1500 kHz. If all's working OK, at the output of the mixer you should have four frequencies: 1500 (original signal) 1955 (oscillator signal - osc. working above the signal freq.) 3455 (sum) 455 (difference) My question is by what process does the mixer produce the 3455 and 455 frequencies. I say it's an add and subtract process, my friend says (via mathematics) it's a multiplication process. Who's right? thanks, Joe W9TXU |
A lot of good stuff here and unfortunately some digression.
It might help here if you think this way. The basic RF or as you say, superhetrodyne mixer, does what it does by instantaneous voltages multiplication of the _instantaneous voltages_ of the two signals. There are many types of these RF mixers, but the one which is modeled with this concept will only give you the two frequencies as Roy's trig identity shows. You can take a spreadsheet and put a sine wave of one freq in one column and another freq in another column (use a pretty small angle step, say 1 degree or less - be careful the degree/radian issue doesn't mess you up) then make a formula in a third column which is A*B. You will see that the resulting "product" column has variations which are not the two input frequencies. It may not me real clear, but if it wiggles faster or slower that the originals, then it is a different frequency. (now I gotta do this so I see what it looks like for my self). However, better yet..... The neatest example of this is to put the exact same frequency in both inputs columns! Then the "wiggles" of the sine wave will be very obvious. There will be no doubt about what is coming out of the multiplication. God, I love spreadsheets. (sorry here, the beginning of a sentence is capitalized). Why does this Mister Wizzard stunt work? Because a circuit is doing something to the voltages present at any given time, and in this case it is a product thing. SO, YES you do get the "Sum" and "Difference" frequencies out, so if you want to call that addition / subtraction while working "in the frequency domain" that's ok with me. However, all this other garbage holds, just the same. If you make a circuit which gives as its output the product of the two input voltages, Roy's formula holds and you get the sum and difference frequencies only. This is what we commonly call a "balanced mixer". The term "balanced" comes from the concept that in this type, if you get the circuit set up or "balanced" just right, the two input signals don't appear at the output and the trig identity holds. I suppose it can be called the ideal type. When you get into what is commonly called "modulation", you still have this type of instantaneous voltage multiplication, but usually, like in a Plate modulated Tube transmitter, it is not so perfect and some of the original input signals get through to the output (though the audio can't make it out to the antenna) and you get carrier (one of the input signals) as well. (I'm not going to get into the 'does the carrier vary in amplitude' or sideband arguments here.) All this talk about many more than the two frequencies is the result of what we call "higher order" non linearities. This is just a way to describe distortion that keeps the original sine waves from being perfect sine waves in a circuit. Also, the sampling talk will just confuse this basic issue, so I advise ignoring it for now. FWIW: the model in my brain can somewhat consider time variant the same as non linearity since you get out something which ain't a simple scaled version of the input... 73, Steve K;9;D:C:I How'm I doin' Roy & Reg? "W3JDR" wrote in message ... Bill, You said: " Any good technician will tell you it's an add and subtract process. Any good engineer will bore you to tears with complicated mathematical analysis. Guess which answer is more useful for your purpose?" Well of course...if you're only interested in what some of what comes out, then it's an 'add and subtract process'. But isn't that our initial definition of what we want a mixer to do? This is circular logic. You're chasing your own tail. The original question was more in the vein of 'by what mechanism does a mixer produce sum and difference frequency components'. The correct answer is that it implements the mathematical product of the two input signals, and that product contains sum and difference frequencies in addition to a host of other frequencies that includes the original frequencies, all their harmonics, and every conceivable product of those frequencies and their harmonics. It's not just a simple 'add and subtract'. It just so happens that we're most interested in the sum and difference, but there is much, much more going on. The "answer that is most useful for the purpose" is not necessarily the most simplistic. Consider the following profound statement from W.E. Deming: "If you can't describe what you are doing as a process, then you don't know what you are doing" Joe W3JDR |
A lot of good stuff here and unfortunately some digression.
It might help here if you think this way. The basic RF or as you say, superhetrodyne mixer, does what it does by instantaneous voltages multiplication of the _instantaneous voltages_ of the two signals. There are many types of these RF mixers, but the one which is modeled with this concept will only give you the two frequencies as Roy's trig identity shows. You can take a spreadsheet and put a sine wave of one freq in one column and another freq in another column (use a pretty small angle step, say 1 degree or less - be careful the degree/radian issue doesn't mess you up) then make a formula in a third column which is A*B. You will see that the resulting "product" column has variations which are not the two input frequencies. It may not me real clear, but if it wiggles faster or slower that the originals, then it is a different frequency. (now I gotta do this so I see what it looks like for my self). However, better yet..... The neatest example of this is to put the exact same frequency in both inputs columns! Then the "wiggles" of the sine wave will be very obvious. There will be no doubt about what is coming out of the multiplication. God, I love spreadsheets. (sorry here, the beginning of a sentence is capitalized). Why does this Mister Wizzard stunt work? Because a circuit is doing something to the voltages present at any given time, and in this case it is a product thing. SO, YES you do get the "Sum" and "Difference" frequencies out, so if you want to call that addition / subtraction while working "in the frequency domain" that's ok with me. However, all this other garbage holds, just the same. If you make a circuit which gives as its output the product of the two input voltages, Roy's formula holds and you get the sum and difference frequencies only. This is what we commonly call a "balanced mixer". The term "balanced" comes from the concept that in this type, if you get the circuit set up or "balanced" just right, the two input signals don't appear at the output and the trig identity holds. I suppose it can be called the ideal type. When you get into what is commonly called "modulation", you still have this type of instantaneous voltage multiplication, but usually, like in a Plate modulated Tube transmitter, it is not so perfect and some of the original input signals get through to the output (though the audio can't make it out to the antenna) and you get carrier (one of the input signals) as well. (I'm not going to get into the 'does the carrier vary in amplitude' or sideband arguments here.) All this talk about many more than the two frequencies is the result of what we call "higher order" non linearities. This is just a way to describe distortion that keeps the original sine waves from being perfect sine waves in a circuit. Also, the sampling talk will just confuse this basic issue, so I advise ignoring it for now. FWIW: the model in my brain can somewhat consider time variant the same as non linearity since you get out something which ain't a simple scaled version of the input... 73, Steve K;9;D:C:I How'm I doin' Roy & Reg? "W3JDR" wrote in message ... Bill, You said: " Any good technician will tell you it's an add and subtract process. Any good engineer will bore you to tears with complicated mathematical analysis. Guess which answer is more useful for your purpose?" Well of course...if you're only interested in what some of what comes out, then it's an 'add and subtract process'. But isn't that our initial definition of what we want a mixer to do? This is circular logic. You're chasing your own tail. The original question was more in the vein of 'by what mechanism does a mixer produce sum and difference frequency components'. The correct answer is that it implements the mathematical product of the two input signals, and that product contains sum and difference frequencies in addition to a host of other frequencies that includes the original frequencies, all their harmonics, and every conceivable product of those frequencies and their harmonics. It's not just a simple 'add and subtract'. It just so happens that we're most interested in the sum and difference, but there is much, much more going on. The "answer that is most useful for the purpose" is not necessarily the most simplistic. Consider the following profound statement from W.E. Deming: "If you can't describe what you are doing as a process, then you don't know what you are doing" Joe W3JDR |
Thanks everyone, in fact I received an e-mail from my friend with
similar trigonometric equations, so I'm absolutely convinced! I now have a slightly better idea of how a superhet mixer functions .... Joe W9TXU |
Thanks everyone, in fact I received an e-mail from my friend with
similar trigonometric equations, so I'm absolutely convinced! I now have a slightly better idea of how a superhet mixer functions .... Joe W9TXU |
Ian:
[snip] "Ian White, G3SEK" wrote in message ... Roy Lewallen wrote:: : Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! : 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) [snip] Mixer, modulator, multiplier, demodulator, detector, switcher, balanced modulator, adder, subtractor, heh, heh.... The term mixer is overused, or... "overloaded" as the computer scientists like to say. Yes indeed, too bad for beginners, but it's part of the mystique of our trade as well, that there are plenty of examples of misuse, misappropriation, and the outright abuse of terms and their meanings in our trade! Keeps gurus in business and nosey outsiders out, as well. :-) Heh, heh... Even within the English speaking community, there is often no consistency of terminology use, for example "tube" versus "valve", etc... British and American use of the term "mixer" in the television production equipment business has further confusing examples of overuse and overlapping meanings. In television production technology the term "mixer" is also used to describe switching and sepcial effects equipment and the terms are applied differently on each side of the Atlantic. What you Brits call a television "mixer" is called a television "switcher" in America, and what's more... the same names are used for the operators of the said mixing/switching equipment. [Grass Valley, Ross, Central Dynamics, etc... are manufacturers of such.] You can often see the equipment operator's names listed opposite the titles Mixer or Switcher on the TV screen when they roll the credits at the end of television shows. And to make things worse, the "function" of an audio "mixer" is again entirely different than a video "mixer", whilst television video mixers often contain integrated audio mixers. Impossible for beginners to figure out what experts are talking about, go figure! -- Peter K1PO Indialantic By-the-Sea, FL |
Ian:
[snip] "Ian White, G3SEK" wrote in message ... Roy Lewallen wrote:: : Part of the confusion is that audio engineers talk about "mixing" where they actually mean adding. Mixing - as RF engineers use the term - is precisely what they don't want! : 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) [snip] Mixer, modulator, multiplier, demodulator, detector, switcher, balanced modulator, adder, subtractor, heh, heh.... The term mixer is overused, or... "overloaded" as the computer scientists like to say. Yes indeed, too bad for beginners, but it's part of the mystique of our trade as well, that there are plenty of examples of misuse, misappropriation, and the outright abuse of terms and their meanings in our trade! Keeps gurus in business and nosey outsiders out, as well. :-) Heh, heh... Even within the English speaking community, there is often no consistency of terminology use, for example "tube" versus "valve", etc... British and American use of the term "mixer" in the television production equipment business has further confusing examples of overuse and overlapping meanings. In television production technology the term "mixer" is also used to describe switching and sepcial effects equipment and the terms are applied differently on each side of the Atlantic. What you Brits call a television "mixer" is called a television "switcher" in America, and what's more... the same names are used for the operators of the said mixing/switching equipment. [Grass Valley, Ross, Central Dynamics, etc... are manufacturers of such.] You can often see the equipment operator's names listed opposite the titles Mixer or Switcher on the TV screen when they roll the credits at the end of television shows. And to make things worse, the "function" of an audio "mixer" is again entirely different than a video "mixer", whilst television video mixers often contain integrated audio mixers. Impossible for beginners to figure out what experts are talking about, go figure! -- Peter K1PO Indialantic By-the-Sea, FL |
Bill Turner wrote:
Correct of course, but as I understand it, the only complication is that there are harmonics present. It is still a case of add and subtract, isn't it? The original question only mentioned the *four* frequencies present in the output, ignoring the harmonics. Unless you can show me otherwise, I stand by my original observation. The math does show otherwise. When we talk about "square-law" and "third-order", we're actually buying into a whole package deal of math-based concepts. Logically, the deal is that we can't use those words *meaningfully* unless we also accept what the math tells us, namely: 1. Each order of distortion is independent of all the other orders. It generates its own individual package of output frequencies. 2. Frequencies that are in the same-order package *must* all be generated together (you can't have one of them without having all the others too). 3. Frequencies that are in different-order packages are totally separate and unconnected. 2f1 and 2f2 are part of the package of 2nd-order products, along with (f1 + f1) and (f1 - f2)... there are four 2nd-order output frequencies, no more and no less. 3f1, 3f2, (2f1 + f2), (2f1 - f2), (f1 + 2f1) and (f2 - 2f1) are all part of the 3rd-order package... there are six 3rd-order output frequencies, no more and no less. A perfect square-law mixer produces only 2nd-order products. 2f1 and 2f2 are present at the output, but they do not "go round again" and mix with the input signals to produce (2f1 + f2) etc. Those 3rd-order products arise *entirely and exclusively* from 3rd-order distortion. That conclusion follows by strict, non-negotiable mathematical logic from the fundamental definition of what "order of distortion" means. As others have said, from the practical engineering point of view, the way you envision mixing products being produced is "purely academic". But that "purely academic" debate is exactly what we're involved in here... so here, it makes all the difference in the world. -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
Bill Turner wrote:
Correct of course, but as I understand it, the only complication is that there are harmonics present. It is still a case of add and subtract, isn't it? The original question only mentioned the *four* frequencies present in the output, ignoring the harmonics. Unless you can show me otherwise, I stand by my original observation. The math does show otherwise. When we talk about "square-law" and "third-order", we're actually buying into a whole package deal of math-based concepts. Logically, the deal is that we can't use those words *meaningfully* unless we also accept what the math tells us, namely: 1. Each order of distortion is independent of all the other orders. It generates its own individual package of output frequencies. 2. Frequencies that are in the same-order package *must* all be generated together (you can't have one of them without having all the others too). 3. Frequencies that are in different-order packages are totally separate and unconnected. 2f1 and 2f2 are part of the package of 2nd-order products, along with (f1 + f1) and (f1 - f2)... there are four 2nd-order output frequencies, no more and no less. 3f1, 3f2, (2f1 + f2), (2f1 - f2), (f1 + 2f1) and (f2 - 2f1) are all part of the 3rd-order package... there are six 3rd-order output frequencies, no more and no less. A perfect square-law mixer produces only 2nd-order products. 2f1 and 2f2 are present at the output, but they do not "go round again" and mix with the input signals to produce (2f1 + f2) etc. Those 3rd-order products arise *entirely and exclusively* from 3rd-order distortion. That conclusion follows by strict, non-negotiable mathematical logic from the fundamental definition of what "order of distortion" means. As others have said, from the practical engineering point of view, the way you envision mixing products being produced is "purely academic". But that "purely academic" debate is exactly what we're involved in here... so here, it makes all the difference in the world. -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
How'm I doin' Roy & Reg?
========================= Steve, you're doing fine. Absolutely no reference to Terman, Kraus, or those 3 gentlemen of 118-radials fame who forgot to measure ground conductivity before going home. ;o) ---- Reg |
How'm I doin' Roy & Reg?
========================= Steve, you're doing fine. Absolutely no reference to Terman, Kraus, or those 3 gentlemen of 118-radials fame who forgot to measure ground conductivity before going home. ;o) ---- Reg |
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