Clipped sum of sinusoids

Let's say there is a sine wave signal, and an amplifier with limited output 
amplitude. To produce maximum output power at fundamental frequency, the 
sinusoid could be clipped into square wave. So far so good.

What if the input is a short burst of a sum of several unrelated sine waves 
with different amplitudes and the goal is to match given amplitude spectrum 
and generate maximum power? How hard would be optimal clipping? What should 
be the relationship of phases and amplitudes at the input of the amplifier?

The result indeed depends on the phases, amplitudes and clipping level. I am 
trying to bruteforce the problem by genetic optimization as I don't see any 
systematic approach. The general problem could be restated as generation of 
the most constant envelope signal with given spectrum. Could you suggest any 
approach or related work?

Vladimir Vassilevsky
DSP and Mixed Signal Consultant
www.abvolt.com

On Tue, 02 Oct 2012 10:26:10 -0500, Vladimir Vassilevsky wrote:

> Let's say there is a sine wave signal, and an amplifier with limited
> output amplitude. To produce maximum output power at fundamental
> frequency, the sinusoid could be clipped into square wave. So far so
> good.
> 
> What if the input is a short burst of a sum of several unrelated sine
> waves with different amplitudes and the goal is to match given amplitude
> spectrum and generate maximum power? How hard would be optimal clipping?
> What should be the relationship of phases and amplitudes at the input of
> the amplifier?
> 
> The result indeed depends on the phases, amplitudes and clipping level.
> I am trying to bruteforce the problem by genetic optimization as I don't
> see any systematic approach. The general problem could be restated as
> generation of the most constant envelope signal with given spectrum.
> Could you suggest any approach or related work?
> 
> Vladimir Vassilevsky DSP and Mixed Signal Consultant www.abvolt.com

There may be something related in the OFDM literature, since shape factor 
is a big deal there.

Probably not, but it may be worth a glance.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

"Tim Wescott" <tim@seemywebsite.please> wrote in message 
news:nL-dnQKgNOcjlvbNnZ2dnUVZ_rKdnZ2d@web-ster.com...
> On Tue, 02 Oct 2012 10:26:10 -0500, Vladimir Vassilevsky wrote:
>
>> Let's say there is a sine wave signal, and an amplifier with limited
>> output amplitude. To produce maximum output power at fundamental
>> frequency, the sinusoid could be clipped into square wave. So far so
>> good.
>>
>> What if the input is a short burst of a sum of several unrelated sine
>> waves with different amplitudes and the goal is to match given amplitude
>> spectrum and generate maximum power? How hard would be optimal clipping?
>> What should be the relationship of phases and amplitudes at the input of
>> the amplifier?
>>
>> The result indeed depends on the phases, amplitudes and clipping level.
>> I am trying to bruteforce the problem by genetic optimization as I don't
>> see any systematic approach. The general problem could be restated as
>> generation of the most constant envelope signal with given spectrum.
>> Could you suggest any approach or related work?
>>

> There may be something related in the OFDM literature, since shape factor
> is a big deal there.
> Probably not, but it may be worth a glance.

I looked through that. The OFDM peak problem is quite different, and they 
don't seem to have many ideas to offer either.

VLV

On 10/2/2012 8:26 AM, Vladimir Vassilevsky wrote:
> Let's say there is a sine wave signal, and an amplifier with limited output
> amplitude. To produce maximum output power at fundamental frequency, the
> sinusoid could be clipped into square wave. So far so good.
>
> What if the input is a short burst of a sum of several unrelated sine waves
> with different amplitudes and the goal is to match given amplitude spectrum
> and generate maximum power? How hard would be optimal clipping? What should
> be the relationship of phases and amplitudes at the input of the amplifier?
>
> The result indeed depends on the phases, amplitudes and clipping level. I am
> trying to bruteforce the problem by genetic optimization as I don't see any
> systematic approach. The general problem could be restated as generation of
> the most constant envelope signal with given spectrum. Could you suggest any
> approach or related work?
>
> Vladimir Vassilevsky
> DSP and Mixed Signal Consultant
> www.abvolt.com
>
>
Vlad,

Well it seems that the problem as stated is underconstrained .. so I'm 
not sure there's an answer.

As I recall, there is a beamformer due to Vic Anderson which clips the 
signals at the outset.  Yet, the spectrum is preserved after having done 
that .. as a general statement.  Probably has something to do with the 
bandwidth one looks at thereafter.  Yet, the notion as I recall was for 
fairly wideband beamformers.  Might this suggest "hard clipping" is the 
answer?  Intuitively it seems that would be the answer anyway.

Fred

>Let's say there is a sine wave signal, and an amplifier with limited
output 
>amplitude. To produce maximum output power at fundamental frequency, the 
>sinusoid could be clipped into square wave. So far so good.
>
>What if the input is a short burst of a sum of several unrelated sine
waves 
>with different amplitudes and the goal is to match given amplitude
spectrum 
>and generate maximum power? How hard would be optimal clipping? What
should 
>be the relationship of phases and amplitudes at the input of the
amplifier?
>
>The result indeed depends on the phases, amplitudes and clipping level. I
am 
>trying to bruteforce the problem by genetic optimization as I don't see
any 
>systematic approach. The general problem could be restated as generation
of 
>the most constant envelope signal with given spectrum. Could you suggest
any 
>approach or related work?
>
>Vladimir Vassilevsky
>DSP and Mixed Signal Consultant
>www.abvolt.com
>
>
>

Peak limiting algorithm rather than brute clipping. Peak limiting sets a
threshold for instantaneous peak power in time domain as ratio to mean
power of composite signal. Then it subtracts a percentage of excess value
above threshold from peak itself and all adjacent signal samples on either
side of peak. Thus you shrink the signal gently whenever peaks occur. You
are only limited by the degree of noise floor rise acceptable to you.

Vladimir Vassilevsky <nospam@nowhere.com> wrote:
> Let's say there is a sine wave signal, and an amplifier with limited output 
> amplitude. To produce maximum output power at fundamental frequency, the 
> sinusoid could be clipped into square wave. So far so good.

OK, so a square wave of a given frequency maximizes the component
sine with that frequency and amplitude limit, I agree.

> What if the input is a short burst of a sum of several unrelated sine waves 
> with different amplitudes and the goal is to match given amplitude spectrum 
> and generate maximum power? How hard would be optimal clipping? What should 
> be the relationship of phases and amplitudes at the input of the amplifier?

My first thought was to separately compute the square waves and add
them. Each has to be scaled by the amplitude of the appropriate sine,
and then the sum rescaled to match the clipping level. 

I believe that works as long as no harmonic component of one adds
to the fundamental of another. It can add to any other harmonic, 
which doesn't affect the result.

> The result indeed depends on the phases, amplitudes and clipping level. I am 
> trying to bruteforce the problem by genetic optimization as I don't see any 
> systematic approach. The general problem could be restated as generation of 
> the most constant envelope signal with given spectrum. Could you suggest any 
> approach or related work?

My other thought is to do the problem backwards. You know what the
result has to be, at least some constraints on the result. 
Then do the inverse transform.

Otherwise, it might be easier to figure out if we knew the
application of the result.

-- glen

On Tue, 02 Oct 2012 10:49:01 -0700, Fred Marshall
<fmarshallxremove_the_x@acm.org> wrote:

>As I recall, there is a beamformer due to Vic Anderson which clips the 
>signals at the outset.  

DIMUS beamformer.

>Yet, the spectrum is preserved after having done 
>that .. as a general statement.  Probably has something to do with the 
>bandwidth one looks at thereafter.  

Also has to do with the fact that many separate versions of the same signals are
clipped, delayed, and added. Correlated signals still add coherently and
uncorrelated signals still add incoherently (more or less), despite the
hard-clipping -- which is what a beamformer is for. 

Vladimir appears to be adding many clipped "unrelated" signals, which may tend
to add like uncorrelated signals.

Greg

On Tuesday, October 2, 2012 5:26:10 PM UTC+2, Vladimir Vassilevsky wrote:
> Let's say there is a sine wave signal, and an amplifier with limited output 
> 
> amplitude. To produce maximum output power at fundamental frequency, the 
> 
> sinusoid could be clipped into square wave. So far so good.
> 
> 
> 
> What if the input is a short burst of a sum of several unrelated sine waves 
> 
> with different amplitudes and the goal is to match given amplitude spectrum 
> 
> and generate maximum power? How hard would be optimal clipping? What should 
> 
> be the relationship of phases and amplitudes at the input of the amplifier?
> 
> 
> 
> The result indeed depends on the phases, amplitudes and clipping level. I am 
> 
> trying to bruteforce the problem by genetic optimization as I don't see any 
> 
> systematic approach. The general problem could be restated as generation of 
> 
> the most constant envelope signal with given spectrum. Could you suggest any 
> 
> approach or related work?
> 
> 
> 
> Vladimir Vassilevsky
> 
> DSP and Mixed Signal Consultant
> 
> www.abvolt.com

GRUMPIDENT!

take a look at voltera kernels.

>> take a look at voltera kernels.

could you provide more information, please?
Volterra kernels for weakly nonlinear systems, business as usual. 
But, hard clipping, I don't see how the pieces fit together.

Somewhere in the distant past I recall a paper about maximizing the range of AM broadcast stations by minimizing the peak to average ratio. I think it worked by dynamically altering the phase response of the audio channel. But I can't find the reference, sorry.  

Bob