Harmonics of clipped sinusoid?

[Jackson Harvey]

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid? That is, if the amplifier output
is X(t) when -1 < X(t) < 1, 1 when X(t) > 1, and -1 when X(t) < -1, and
I drive it with X(t) = A*sin(2*pi*t), I will get b_1*sin(2*pi*t) +
b_3*sin(2*pi*3*t) + b_5*sin*(2*pi*5*t) + ... + b_n*sin(2*pi*n*t) I
would like to find b_1, b_3,...b_n in terms of A. I get stuck in the
math (although if anyone can simplify 2/n*cos(n*arcsin(1/A)), and maybe
give a simpler form of A*arcsin(1/A), I could finish the math).

BTW, this is not homework. I wanted to evaluate the products that would
be generated in an ideal switching mixer based on the amplitude of the
LO drive to that mixer. The mixer used a differential pair for current
steering, which can be approximated to the first order by an ideal
clipping amplifier.

Thanks in advance,

Jackson Harvey

 

[John Woodgate]

I read in sci.electronics.design that Jackson Harvey

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid? That is, if the amplifier output
is X(t) when -1 < X(t) < 1, 1 when X(t) > 1, and -1 when X(t) < -1, and
I drive it with X(t) = A*sin(2*pi*t), I will get b_1*sin(2*pi*t) +
b_3*sin(2*pi*3*t) + b_5*sin*(2*pi*5*t) + ... + b_n*sin(2*pi*n*t) I
would like to find b_1, b_3,...b_n in terms of A. I get stuck in the
math (although if anyone can simplify 2/n*cos(n*arcsin(1/A)), and maybe
give a simpler form of A*arcsin(1/A), I could finish the math).

There is a standard result for a 'sine cap', which is the bit that's
clipped off. So you just subtract that from the original sine wave.

I have the standard result only in a very general form, which is very
difficult to render in ASCII. I will post it if you don't get a more
accessible form from someone else.

 

[Tim Wescott]

I read in sci.electronics.design that Jackson Harvey
'Harmonics of clipped sinusoid?', on Fri, 20 Aug 2004:




There is a standard result for a 'sine cap', which is the bit that's
clipped off. So you just subtract that from the original sine wave.

I have the standard result only in a very general form, which is very
difficult to render in ASCII. I will post it if you don't get a more
accessible form from someone else.

sin(arcsin(1/A))=1/A

for x = sin(y) cos(y) = sqrt(1 - x^2).

So cos(arcsin(1/A)) = sqrt(1 - 1/A^2).

cos(n*y) = 2*cos(n-1)*y*cos(y) - cos(n-2)*y

cos(n*arcsin(1/A)) is left as an exercise to the reader, 'cause I'm lazy.

Or find that 'sine cap' formula, and you won't be caught up in your
clerical errors plus mine.

 

[Norm Dresner]

I read in sci.electronics.design that Jackson Harvey


There is a standard result for a 'sine cap', which is the bit that's
clipped off. So you just subtract that from the original sine wave.

I have the standard result only in a very general form, which is very
difficult to render in ASCII. I will post it if you don't get a more
accessible form from someone else.

Do you have any information on the harmonic components/fourier
coefficients/THD of approximating a sine wave with a "perfect" DAC of N-bits
as a function of N?

Norm

 

[John Woodgate]

Do you have any information on the harmonic components/fourier
coefficients/THD of approximating a sine wave with a "perfect" DAC of N-
bits as a function of N?

No, sorry. A Google for 'magic sinewaves' might help.

 

[James Meyer]

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid?

Jackson Harvey

Doesn't the output of an ideal clipper approach a square wave with
sufficient accuracy?

Jim

 

[Jackson Harvey]

Doesn't the output of an ideal clipper approach a square wave with
sufficient accuracy?

Jim

Certainly, as input amplitude gets large. And, of course, as amplitude
gets small you just get the sine wave you put in. There is a smooth
transition in between, but I do not currently have a formula for the
harmonics in this in-between area.

 

[Helmut Sennewald]

Do you have any information on the harmonic components/fourier
coefficients/THD of approximating a sine wave with a "perfect" DAC of N-bits
as a function of N?

Hello Norm,
I have some results from simulation of an ideal N-bit DAC with LTSPICE.
Maybe the spurs can be further improved slighly by moving the code
change a lttle bit.

The magic formulas from a fit:
------------------------------

Spurs_dB: -8.5dB*N Fit from simulation with LTSPICE

THD_dB: -6dB*N -1.9dB SNR 6dB*N + 1.76dB

The last is is the formula you find in DDS articles for signal/noise ratio.
I found that THD is very close to this equation.
The same result would be achieved with a DDS sine generator as long
as the relationship Fsignal/Fclock is very low and no filter is in use.
The highest spurs are concentrated at very high harmonic numbers.

Results from LTSPICE when codes from 1 ... 2**N -1 were used.
Example: N=8 codes: 1 .. 255 center=128, code 0 not used

N-bit Spur[dB]
3 -25
4 -36
5 -43
6 -52
7 -60.5
8 -68
9 -77
10 -84
12 -102

N=8bit: Total THD 1.1mVrms with a signal of 350mVrms agrees well
with DDS formula for SNR.


Best regards,
Helmut

 

[The Phantom]

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid? That is, if the amplifier output
is X(t) when -1 < X(t) < 1, 1 when X(t) > 1, and -1 when X(t) < -1, and
I drive it with X(t) = A*sin(2*pi*t), I will get b_1*sin(2*pi*t) +
b_3*sin(2*pi*3*t) + b_5*sin*(2*pi*5*t) + ... + b_n*sin(2*pi*n*t) I
would like to find b_1, b_3,...b_n in terms of A. I get stuck in the
math (although if anyone can simplify 2/n*cos(n*arcsin(1/A)), and maybe
give a simpler form of A*arcsin(1/A), I could finish the math).

BTW, this is not homework. I wanted to evaluate the products that would
be generated in an ideal switching mixer based on the amplitude of the
LO drive to that mixer. The mixer used a differential pair for current
steering, which can be approximated to the first order by an ideal
clipping amplifier.

Thanks in advance,

Jackson Harvey

How many harmonics do you care about?

 

[Fred Bloggs]

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid? That is, if the amplifier output
is X(t) when -1 < X(t) < 1, 1 when X(t) > 1, and -1 when X(t) < -1, and
I drive it with X(t) = A*sin(2*pi*t), I will get b_1*sin(2*pi*t) +
b_3*sin(2*pi*3*t) + b_5*sin*(2*pi*5*t) + ... + b_n*sin(2*pi*n*t) I
would like to find b_1, b_3,...b_n in terms of A. I get stuck in the
math (although if anyone can simplify 2/n*cos(n*arcsin(1/A)), and maybe
give a simpler form of A*arcsin(1/A), I could finish the math).

You drop the insistence on making A the parameter and use 0<x<=pi/2 as
the amplitude parameter where A=1/sin(x), specify your harmonic
amplitudes in terms of x, you get a bunch of cos(nx) terms. For A<1, all
harmonics other than fundamental are zero magnitude- so the x-variable
covers it.

 

[The Phantom]

Has anyone seen the math to calculate the amplitudes of the fundamental
components and harmonic components of the output of an ideal clipping
amplifier whose input is a sinusoid? That is, if the amplifier output
is X(t) when -1 < X(t) < 1, 1 when X(t) > 1, and -1 when X(t) < -1, and
I drive it with X(t) = A*sin(2*pi*t), I will get b_1*sin(2*pi*t) +
b_3*sin(2*pi*3*t) + b_5*sin*(2*pi*5*t) + ... + b_n*sin(2*pi*n*t) I
would like to find b_1, b_3,...b_n in terms of A. I get stuck in the
math (although if anyone can simplify 2/n*cos(n*arcsin(1/A)), and maybe
give a simpler form of A*arcsin(1/A), I could finish the math).

BTW, this is not homework. I wanted to evaluate the products that would
be generated in an ideal switching mixer based on the amplitude of the
LO drive to that mixer. The mixer used a differential pair for current
steering, which can be approximated to the first order by an ideal
clipping amplifier.

Thanks in advance,

Have a look on alt.binaries.schematics.electronic for a solution.

 

[Jackson Harvey]

Have a look on alt.binaries.schematics.electronic for a solution.

That's great! Thanks very much.

Jackson Harvey

 

[The Phantom]

<snip"

Do you have any information on the harmonic components/fourier
coefficients/THD of approximating a sine wave with a "perfect" DAC of N-bits
as a function of N?

Norm

You posted a similar query on May 16, and I posted a response on
alt.binaries.schematics.electronic. I've added some new curves and
re-posted it over there. Does it answer your needs?

 

[Norm Dresner]

<snip"


You posted a similar query on May 16, and I posted a response on
alt.binaries.schematics.electronic. I've added some new curves and
re-posted it over there. Does it answer your needs?

Absolutely. I hadn't seen the previous (May) post because my ISP wasn't
carrying that NG then but I've gotten it now.

Someday I've got to either get a copy of Mathematica [excruciatingly
expensive if you're not an academic when it just costs two limbs] or really
learn how to use either MathCad or Matlab. If I only had the time ...

Thanks
Norm