Is it possible to generate harmonic distortion analytically?
in [DSP]
|
From: robert bristow-johnson on 13 Mar 2010 01:01 On Mar 13, 12:15 am, Les Cargill <lcargil...(a)comcast.net> wrote: > robert bristow-johnson wrote: > > > we know that the periodic "dirac comb" function (with period 1/f0) in > > the time domain is another dirac comb in the frequency domain, but i'm > > not gonna prove it. > > > +inf > > d(t) = 1/f0 * SUM{ delta(t - n/f0) } > > n=-inf > > > has for a Fourier Transform: > > > +inf > > D(f) = SUM{ delta(f - k*f0) } > > k=-inf > > > f0 is the fundamental frequency of this periodic function. in the > > frequency domain, there is a spike (a dirac impulse) of height=1 > > spaced apart from its neighbor by f0. > > > now we run that impulse train, d(t), through an ideal brickwall low- > > pass filter with frequency response: > > > H(f) = 1/(2N) * rect( f/((2N+1)*f0) ) > > > where > > > { 1 |u| < 1/2 > > rect(u) = { > > { 0 |u| > 1/2 > > > the rect() function is centered at u=0 has height of 1 and width of > > 1. rect(1/2) usually is defined as 1/2. > > > H(f) has gain of 1 in the passband and the passband width is (2N+1)*f0 > > (encompassing both positive and negative f). that means there are 2N > > +1 spikes in the frequency domain that are passed (with gain 1/(2N)) > > and all the rest are killed. one dirac spike at DC, N spikes in > > positive frequency and N spikes in negative frequency. > > > the output spectrum is > > > N > > Y(f) = H(f) * D(f) = 1/(2N) * SUM{ delta(f - k*f0) } > > k=-N > > > the impulse response of that filter is: > > > h(t) = 1/(2N) * (2N+1)*f0 * sinc((2N+1)*f0*t) > > Sorry, but what is the basis for this? It's a deconvolution, but > what's the name of the mechanism that justifies this transform? > > I am, uh, not up to deconvolution as an art form just yet. it's not about deconvolutions. it's about a Fourier Transform pair: if x(t) = rect(t) then X(f) = sinc(f) that can be figgered out by use of an integral. there's the use of the "duality" theorem to switch f and t around, and then there is some scaling and linearity involved and that is the basis for the impulse response, h(t). this is what they made me learn when i took my first class in Communications Systems as an undergrad (not meant to be a criticism, just to point out that it's not really rocket science). that transform pair with d(t) and D(f) is another. r b-j
From: Les Cargill on 13 Mar 2010 11:55 robert bristow-johnson wrote: > On Mar 13, 12:15 am, Les Cargill <lcargil...(a)comcast.net> wrote: >> robert bristow-johnson wrote: >> >>> we know that the periodic "dirac comb" function (with period 1/f0) in >>> the time domain is another dirac comb in the frequency domain, but i'm >>> not gonna prove it. >>> +inf >>> d(t) = 1/f0 * SUM{ delta(t - n/f0) } >>> n=-inf >>> has for a Fourier Transform: >>> +inf >>> D(f) = SUM{ delta(f - k*f0) } >>> k=-inf >>> f0 is the fundamental frequency of this periodic function. in the >>> frequency domain, there is a spike (a dirac impulse) of height=1 >>> spaced apart from its neighbor by f0. >>> now we run that impulse train, d(t), through an ideal brickwall low- >>> pass filter with frequency response: >>> H(f) = 1/(2N) * rect( f/((2N+1)*f0) ) >>> where >>> { 1 |u| < 1/2 >>> rect(u) = { >>> { 0 |u| > 1/2 >>> the rect() function is centered at u=0 has height of 1 and width of >>> 1. rect(1/2) usually is defined as 1/2. >>> H(f) has gain of 1 in the passband and the passband width is (2N+1)*f0 >>> (encompassing both positive and negative f). that means there are 2N >>> +1 spikes in the frequency domain that are passed (with gain 1/(2N)) >>> and all the rest are killed. one dirac spike at DC, N spikes in >>> positive frequency and N spikes in negative frequency. >>> the output spectrum is >>> N >>> Y(f) = H(f) * D(f) = 1/(2N) * SUM{ delta(f - k*f0) } >>> k=-N >>> the impulse response of that filter is: >>> h(t) = 1/(2N) * (2N+1)*f0 * sinc((2N+1)*f0*t) >> Sorry, but what is the basis for this? It's a deconvolution, but >> what's the name of the mechanism that justifies this transform? >> >> I am, uh, not up to deconvolution as an art form just yet. > > it's not about deconvolutions. it's about a Fourier Transform pair: > > if x(t) = rect(t) > > then X(f) = sinc(f) > Ah! Gotcha. > that can be figgered out by use of an integral. there's the use of > the "duality" theorem to switch f and t around, and then there is some > scaling and linearity involved and that is the basis for the impulse > response, h(t). this is what they made me learn when i took my first > class in Communications Systems as an undergrad (not meant to be a > criticism, just to point out that it's not really rocket science). Understood. > that transform pair with d(t) and D(f) is another. > > r b-j -- Les Cargill
From: jim on 14 Mar 2010 09:53
robert bristow-johnson wrote: > it's not about deconvolutions. it's about a Fourier Transform pair: > > if x(t) = rect(t) > > then X(f) = sinc(f) > If you are dealing with discrete data. this would be if x[t] = rect(t) then X[x] = sin(N*pi*x)/N*sin(pi*x) where square brackets indicate you are working with discrete samples. Sin(Nx)/sin(x) is often called a periodic sinc (Dirichlet). That waveform can be generated as a sum of of cosine functions - that may be what Les was trying to say. -jim |