# Circuit to retain only the positive frequency components in a signal?

• sanjaysan
In summary, a negative frequency component is redundant and can be eliminated by generating a one-sided spectrum without losing generality.
sanjaysan
Hi,
Is there any method(circuit) to retain only the positive frequency components of a signal?

What is a positive frequency component?

when a signal is represented in frequency domain, it has both positive and negative frequency components. example:- cos(2*pi*f*t) in freq domain has 0.5*d(t) at +f and -f.
I am looking to eliminate the component at -f.
(d(t)=delta function)

This reference is in the frequency domain. You will only see all of the negative frequency if you mix the signal with a "carrier" frequency that is at the highest frequency of the signal. A bandpass filter that is to the "right" of the carrier frequency will let the positive side of the signal through while impeding the negative side of the signal.

2milehi said:
This reference is in the frequency domain. You will only see all of the negative frequency if you mix the signal with a "carrier" frequency that is at the highest frequency of the signal. A bandpass filter that is to the "right" of the carrier frequency will let the positive side of the signal through while impeding the negative side of the signal.

I understand what you are trying to say. But, what i was looking for is a means of eliminating the original negative frequency component in the modulating signal without having to shift it by the frequency of the carrier. In other words, i don't want any frequency components towards the left half of the y-axis in the frequency spectrum.

The only time you will see the negative frequencies (in time domain) is when you mix the signal with a carrier. I am not sure of this, but do filters affect only the positive frequencies or do they have the same mirrored profile in the negative side of the frequency domain. My hunch is that filters do have an even symmetry wrt the y-axis in the frequency domain. And if that is the case, then you would have to mix the signal to bring out negative part of it.

Why the need to eliminate the negative part of a signal?

Yes, filters do have a mirrored profile about y-axis in frequency domain, hence filtering out the negative frequency component is not possible. Anyway, why i had this question was that i noticed that any periodic signal has symmetric mirrored profile about the y-axis in f domain. This, i think is a redundancy and i am trying to find a means of eliminating one half of the freq. spectrum(negative part).
On a related note, do the negative frequencies constitute half of the total signal power?

Look into methods of generating a single sideband signal. There is a way to do it without brute force filtering.

sanjaysan, the negative frequency components are redundant, in a sense. Consider your time domain signal, $cos(2\pi \omega t)$. The angular frequency, $\omega$, could be either positive or negative, and the resulting wave would look the same in the time domain. That ambiguity leads to the two-sided, symmetric spectrum.

You can move to a one-sided spectrum if you wish, with no loss of generality, but that's just a mathematical trick. You don't need to design any real, physical device to discard the negative frequencies; they're all in your head from the beginning!

- Warren

Warren, Are you saying there is no physical significance attached to negative frequencies?

Which energy is defined as negative frequency and which is defined as positive depends on the timing (phase) of the FFT function itself, and is not some independent attribute of a signal. When thinking theoretically, you're in the habit of defining some t=0, which will define what is negative and what is positive. Negative and positive frequencies have physical significance only in a process that already includes a FFT function with a defined timing (a defined t=0 which has some physical meaning, like when the FFT circuit is reset). In an actual circuit you would need to "reset" the FFT process at some point in order to define a t=0, if you want frequency polarity to have physical meaning. And once you do all that, you've got a frequency-domain signal that makes it easy to remove the negative frequencies, anyway. Then do a IFFT. There's no shortcut to doing a FT, since FT, by definition, means "getting the frequency components of a time domain signal" (although admittedly how one gets the frequency components is up for efficiency improvements). Certainly there would be a minor (probably insignificant) simplification of a standard FFT circuit/algorithm when all you wanted was the positive frequencies.

Last edited:
sanjaysan said:
Warren, Are you saying there is no physical significance attached to negative frequencies?

"Physical significance"?
The spectrum is just a mathematical way of describing things. Sometimes, the maths suggests the presence of signals but you can't necessarily measure them. This is only like saying that you want a 'Physical Significance' for one of the roots of a (mathematical) equation of motion which doesn't fit the physical problem. You just have to pick the more 'suitable' result.
Try this link to see a thread which touches on just this topic.

sophiecentaur said:
"Physical significance"?
The spectrum is just a mathematical way of describing things. Sometimes, the maths suggests the presence of signals but you can't necessarily measure them. This is only like saying that you want a 'Physical Significance' for one of the roots of a (mathematical) equation of motion which doesn't fit the physical problem. You just have to pick the more 'suitable' result.
Try this link to see a thread which touches on just this topic.
Sophie, I get that we have been 'picking the more suitable result' for analysing the signal. But, I recently read a paper titled "Negative frequency communication" which says that half of the frequency(negetive) is being wasted and it does have a physical meaning. You can read more about it here-
http://arxiv.org/abs/1012.1403
so is it just a mathematical eventuality or is there more meaning to it??

I can only read the abstract but that paper seems to be drawing conclusions based on ancient forms of modulation (dsbam, for instance). For decades, modulation systems have been used which do not have a 'redundant' half to their spectrum. SSB has been used since before WW2, afaik and that was 'as crude as you could imagine'.
People have used 'independent sideband' transmissions and suppressed the carrier.

I may be doing the author a disservice here but it certainly looks like that to me. I would need access to the whole paper but I can't seem to find it.

sophiecentaur said:
I can only read the abstract but that paper seems to be drawing conclusions based on ancient forms of modulation (dsbam, for instance). For decades, modulation systems have been used which do not have a 'redundant' half to their spectrum. SSB has been used since before WW2, afaik and that was 'as crude as you could imagine'.
People have used 'independent sideband' transmissions and suppressed the carrier.

I may be doing the author a disservice here but it certainly looks like that to me. I would need access to the whole paper but I can't seem to find it.
I've read the whole paper. Its not about SSB modulation. The author specifically talks about using the negetive frequency components for communication. Heres the link to the paper-
http://arxiv.org/pdf/1012.1403v5.pdf
Any feedback would be appreciated.

Well there is no such thing as negative frequency. In real life. That doesn't make sense.

I only think math behind the Fourier transform, and generally math tools are made in that way so that we do operate with negative frequencies.

Cosine is represented with 2 deltas right? (amplitude spectra)

Eliminate one of them, and you no longer have a cosine...

Specifically let's consider this.

$F (cos(\omega _0 t)) = \pi\left[\delta (\omega -\omega _0)+\delta (\omega +\omega _0)\right]$

Eliminate the negative frequency and you get:

$\pi\left[ \delta (\omega -\omega _0)\right]$

Inverse Fourier transform of this is no longer a cosine. Its complex exponential:

$F^{-1}(\pi\left[ \delta (\omega -\omega _0)\right])=\frac{1}{2}\cdot e^{j\omega _0 t}$

So its not even a real function anymore.

You can go with using a complex exponential as your carrier.
Design a system, which transfers in parallel a Sine and a Cosine.

Last edited:
Looking briefly at the paper I see he describes his chosen form of modulation in terms of multiplying a carrier wave by a modulating signal. Depending upon what the form of the modulating signal (depending upon any constant term) this will produce easily-demodulated AM, Suppressed Carrier AM or something in between. It's clear that this will have redundant frequency components (his 'negative' components). It seems that he is proposing a complex system of modulation to eliminate one of those sidebands. This, again has been done (A Marconi Patent, I believe - or it may be Siemens) with high powered HF transmitters as an efficient way of producing SSB (just one of the sidebands). The modulation consists of a combination of AM and Phase Modulation, in which the sidebands on one side cancel.

If you consider the actual Power involved in his proposed system, if it were anything other than some version of this, then (up to) 100% excess power would have to come from somewhere in the transmitter if information were to be available at the other end of the link by way of this extra available channel.
Using two polarisations for two transmitting channels is, again, nothing new. Every TV transmitter network uses polarisation diversity in its service planning.

This seems to me like a good bit of theoretical bookwork that has been done in a bit of a vacuum. He cites no RF Engineering references to put the work in a real world context and I wonder whether he is actually aware that he may well be re-inventing a wheel.

http://pokit.org/get/1ce09bbe329cdfd1847282b373f7d3ac.jpg

Method using sine and cosine as carrier.

http://pokit.org/get/41a7f4cf9f65ff190857c955c0b3712d.jpg

Frequency spectra.

http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/lecture-13-continuous-time-modulation/MITRES_6_007S11_lec13.pdf

Last edited by a moderator:
That would work fine at low power but I think the (200kW??) transmitter I have seen used just one transmitting valve and achieved the result by AM and PM in a single unit.

Hang on a minute. How would you combine your two signals losslessly from two high power amplifiers? I think that could be a problem.

I only know the theory behind it, I never designed one :D

Bassalisk said:
Well there is no such thing as negative frequency. In real life. That doesn't make sense.

I only think math behind the Fourier transform, and generally math tools are made in that way so that we do operate with negative frequencies.

Cosine is represented with 2 deltas right? (amplitude spectra)

Eliminate one of them, and you no longer have a cosine...

Specifically let's consider this.

$F (cos(\omega _0 t)) = \pi\left[\delta (\omega -\omega _0)+\delta (\omega +\omega _0)\right]$

Eliminate the negative frequency and you get:

$\pi\left[ \delta (\omega -\omega _0)\right]$

Inverse Fourier transform of this is no longer a cosine. Its complex exponential:

$F^{-1}(\pi\left[ \delta (\omega -\omega _0)\right])=\frac{1}{2}\cdot e^{j\omega _0 t}$

So its not even a real function anymore.

You can go with using a complex exponential as your carrier.
Design a system, which transfers in parallel a Sine and a Cosine.
Bassalisk, How do i go about generating the complex exponential e^(jwt)? You said a system which transfers sine and cosine waves in parallel. Do you mean sin(wt)+cos(wt)? shouldn't it be cos(wt)+j sin (wt)?

sanjaysan said:
Bassalisk, How do i go about generating the complex exponential e^(jwt)? You said a system which transfers sine and cosine waves in parallel. Do you mean sin(wt)+cos(wt)? shouldn't it be cos(wt)+j sin (wt)?

There wouldn't be imaginary current or potential in an actual circuit. The circuit depicted carries two separate products with sine and cosine waves respectively, to represent multiplying by a complex exponential.

You should know that real signals have Hermitian symmetry in the frequency domain. The component at a negative frequency is the complex conjugate of the component at the corresponding positive frequency.

This has to do with the symmetry of cosine and the "odd" symmetry of sine. Consider what happens if we make the frequency negative.

cos(ωt) = cos(-ωt)

but

sin(ωt) = -sin(-ωt)Anyway, removing the negative frequencies seemingly would imply two outputs instead of one, based on Hermitian symmetry requirement for the frequency domain of real signals.

Last edited:
sanjaysan said:
Bassalisk, How do i go about generating the complex exponential e^(jwt)? You said a system which transfers sine and cosine waves in parallel. Do you mean sin(wt)+cos(wt)? shouldn't it be cos(wt)+j sin (wt)?

Well, imaginary numbers are a pair of real numbers. So that system that I posted in the post above, explains it nicely.

When you go deeper into Signals and Systems, these things are pretty easy to understand :D

I think we are confusing real, time varying signals with a convenient mathematical representation of them. Neither the exponential notation or the 'cis' notation are any more than models, and 'half' of that model is not relevant to the real world. afaik, one should really prefix the final result of any 'complex' jiggery pokery with the words "The Real Part Of . . . " if you want to get a proper answer.

I think you can do the analysis (albeit in a more lumpy way) without using i at all.

sophiecentaur said:
I think we are confusing real, time varying signals with a convenient mathematical representation of them. Neither the exponential notation or the 'cis' notation are any more than models, and 'half' of that model is not relevant to the real world. afaik, one should really prefix the final result of any 'complex' jiggery pokery with the words "The Real Part Of . . . " if you want to get a proper answer.

Well put.

That's where that paper on "negative Frequencies in Modulation" seems to be skating on thin ice.

sophiecentaur said:
That's where that paper on "negative Frequencies in Modulation" seems to be skating on thin ice.

Yes its somewhat like talking about negative time. But nevertheless, this thread was really interesting!

Bassalisk said:
Well, imaginary numbers are a pair of real numbers. So that system that I posted in the post above, explains it nicely.

When you go deeper into Signals and Systems, these things are pretty easy to understand :D
So, is x(t)cos(wt)+y(t)sin(wt) the required complex exponential modulated signal?

If I have a room of 12 square metres area and I want a carpet for it, I would choose one with dimensions 3m by 4m. This would be a REAL carpet that had Real dimensions, made of Real Wool. The Maths would tell me that a carpet with dimension -3m by -4m would also do the job. Only a loony would go out to look for one of those in a shop.

Why look for anything more significant when some Maths suggests that a Negative Frequency component could exist for a signal? At the beginning of the thread there was a question about the existence of Power in this 'Mirror' signal. Clearly not. A square wave oscillator takes power from its (DC) power supply (real, measurable Joules from a battery). This Power is exactly the same as the power in the square wave - as you can see by heating up an element or by integration. There is no other power in the system. The 'negative frequency' component is just an artifact of the Maths - just like the negatively dimensioned carpet. The same applies to a sinewave.

So what about the two sidebands which are generated in AM? They are at (absolute) positive frequencies and carry energy - along with the carrier and, if you add up the three Powers, you get the value of Power which the power supply delivers (less a measurable / calculable factor due to the efficiency of the modulator. The power in each sideband can be used to carry other information and the power of the carrier can be reduced to zero - giving you two SSB transmissions, each of which has the same SNR as the original dsbam signal, half the signal spectrum occupancy and may save the power of a carrier, depending on how you actually produce the ssb signal.

When you draw diagrams of signal spectra and show them shifting around and being filtered, there is no explicit mention of the Power involved. So the diagram proves nothing about the existence or otherwise of 'components'.
I, personally, was a bit disappointed with that MIT Movie. It would be very easy to get some wrong messages from it, I think. OK, as far as it went but strictly an undergraduate treatment of the topic, I should say, aimed at getting predictable results from a comms system.

chroot said:
sanjaysan, the negative frequency components are redundant, in a sense. Consider your time domain signal, $cos(2\pi \omega t)$. The angular frequency, $\omega$, could be either positive or negative, and the resulting wave would look the same in the time domain. That ambiguity leads to the two-sided, symmetric spectrum.

You can move to a one-sided spectrum if you wish, with no loss of generality, but that's just a mathematical trick. You don't need to design any real, physical device to discard the negative frequencies; they're all in your head from the beginning!

- Warren

Then how do you explain the recovery of baseband signal from single sided passband signal. Suppose we have only upper sideband of a signal then in recovery of message signal the mirrored band of the signal contributes to form the spectrum of the message signal. What do you think happens physically here...

Look at the block diagram of an ssb receiver. A local oscillator at the original carrier frequency will mix with the sideband and produce a baseband signal - and other mixing products, of course but they will be at non-baseband frequencies and their power level is not relevant.

If you're concerned about the SNR of the demodulated signal then it would be 3dB lower than when both sidebands are demodulated because the noise bandwidth would be half but the demodulated signal would be half the level - giving 3dB net loss. BUT there was 3dB less power transmitted for a start so there would is overall disadvantage (as long as the transmitter can be made efficient.

It is important to get the Maths and the Physical World reconciled properly. Like I said earlier. All the answers to the calculations should start off with "The real part of".

## 1. How does a circuit retain only the positive frequency components in a signal?

A circuit can retain only the positive frequency components in a signal by using a high-pass filter. This type of filter allows high-frequency signals to pass through while blocking low-frequency signals. By adjusting the cutoff frequency of the filter, only the positive frequency components of the signal will be allowed to pass through.

## 2. What is the purpose of retaining only the positive frequency components in a signal?

The purpose of retaining only the positive frequency components in a signal is to remove any unwanted low-frequency noise or interference. This can improve the quality and clarity of the signal and make it easier to analyze or use for a specific purpose.

## 3. Can a circuit retain only the positive frequency components without affecting the overall signal?

Yes, a circuit can retain only the positive frequency components without affecting the overall signal by using a high-pass filter with a sharp cutoff. This will ensure that only the desired frequency components are removed while leaving the rest of the signal intact.

## 4. What types of signals can benefit from a circuit that retains only the positive frequency components?

A circuit that retains only the positive frequency components can benefit any type of signal, but it is particularly useful for audio signals. By removing low-frequency noise, the audio signal can be clearer and easier to understand. It can also be beneficial for signals in communication systems, scientific experiments, and electronic devices.

## 5. Are there any limitations to using a circuit to retain only the positive frequency components in a signal?

One limitation of using a circuit to retain only the positive frequency components in a signal is that it can only remove low-frequency noise. If the signal is affected by high-frequency noise, a different type of filter, such as a low-pass filter, may be needed. Additionally, the cutoff frequency of the high-pass filter must be carefully selected to avoid removing desired frequency components from the signal.

• Electrical Engineering
Replies
48
Views
2K
• Electrical Engineering
Replies
8
Views
1K
• Electrical Engineering
Replies
17
Views
1K
• Electrical Engineering
Replies
13
Views
1K
• Electrical Engineering
Replies
4
Views
772
• Electrical Engineering
Replies
10
Views
1K
• Electrical Engineering
Replies
30
Views
2K
• Electrical Engineering
Replies
10
Views
2K
• Electrical Engineering
Replies
2
Views
1K
• Electrical Engineering
Replies
12
Views
1K