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negative frequency and Hilbert transform

Started by fisico30 February 6, 2009
hello forum,

I might need some help understanding the usefulness of the complex
analytic signal.

What we measure are real valued signals whose Fourier transform can be
one-sided, or, if we used complex sinusoids, two-sided and symmetric. Then
the real signal is made of positive and negative frequencies in equal
amount. The negative sinusoids dont really a physical meaning, I guess.

Complex analytic signal: it transforms a real signal into a complex signal
with only the positive part of the frequency.
What do we gain? We have been using the double sided spectrum taking
advantage of the complex sinusoids. We could have used simple trigonometric
sines and cosine to get the spectrum instead. The spectrum would have been
positive and one-sided from the beginning. What am i missing?

thanks for any clarification
fisico30 


On Feb 6, 10:25&#2013266080;pm, "fisico30" <marcoscipio...@gmail.com> wrote:
> > What we measure are real valued signals whose Fourier transform can be > one-sided, or, if we used complex sinusoids, two-sided and symmetric.
i think it's the opposite. a real signal has Fourier Transform that is 2-sided with conjugate symmetry. the analytic signal, which is the original real x(t) in the real part and the Hilbert transform of x(t), which we'll call q(t), in the imaginary part. this is the "analytic signal" of x(t) is a(t) = x(t) + j*q(t) where q(t) = Hilbert{ x(t) } the F.T. of this analytic signal is one sided, the negative frequency side is all 0 and the positives are simply twice the amplitude of the same components in the original x(t). you get this be noting that the Hilbert Transformer has frequency response of: H(f) = -j * sgn(f) where sgn(f) = +1 for f>0 and -1 for f<0 (0 for f=0) then A(f) = X(f) + j*Q(f) = X(f) + j*( (-j*sgn(f) * X(f) ) A(f) = X(f) * (1 + sgn(f)) which is A(f) = 2*X(f) for f>0 and A(f) = 0 for f<0
> Then > the real signal is made of positive and negative frequencies in equal > amount. The negative sinusoids dont really a physical meaning, I guess.
they have mathematical meaning. negative frequencies coming out of real sinusoidal functions are not physical perceptible. but e^(- j*w*t) is not the same as e^(+j*w*t). actually, *qualitatively* they are exactly the same, because +j and -j both have equal claim to squaring to -1, but we treat them as not equal mathematically. we treat -j and +j as negatives of each other and not zero. that means they are different.
> Complex analytic signal: it transforms a real signal into a complex signal > with only the positive part of the frequency. > What do we gain?
some mathematical convenience. one of the first DSP algs i ever did was a frequency shifter (not a pitch shifter). essentially it was s'pose to take the positive spectrum and scoot it up or down any specified amount (the negative half scoots down or up, respectively, the same amount). the result is real (the spectrum remains two-sided and conjugate symmetrical), but the most straight-forward way to visualize how this is done is with analytic signals and a multiplier of complex numbers.
> We have been using the double sided spectrum taking > advantage of the complex sinusoids. We could have used simple trigonometric > sines and cosine to get the spectrum instead. The spectrum would have been > positive and one-sided from the beginning. What am i missing?
the Gospel of Euler. it is *much* easier to deal with exponential functions than sinusoidal. they both differentiate and delay and remain the same exponential or sinusoidal form but when exponentials remain the same form, they are no different than a (possibly complex) scaler of the exponential going in. if you differentiate or delay a sinusoid, you have to worry about shifting the sinusoidal form in time. but with an exponential, shifting in time is the same as scaling it. what this allows you to do is treat a linear, time-invariant system as this operation that has exponential functions as their eigenfunction. that means exponential going in means an identical exponential comes out but is scaled a little (the magnitude of that scaler is the gain of the LTI system and the angle of the scaler is the phase-shift of the system). now, it's so sad that we can't do that with sinusoids but, lo, our ass is saved by Leonard Euler and the Truth He has Bestowed upon us: e^(j*theta) = cos(theta) + j*sin(theta) that fact saves our butts a lot of work. r b-j (disciple of Euler)
On Feb 6, 7:25&#2013266080;pm, "fisico30" <marcoscipio...@gmail.com> wrote:
> hello forum, > > I might need some help understanding the usefulness of the complex > analytic signal. > > What we measure are real valued signals whose Fourier transform can be > one-sided, or, if we used complex sinusoids, two-sided and symmetric. Then > the real signal is made of positive and negative frequencies in equal > amount. The negative sinusoids dont really a physical meaning, I guess. > > Complex analytic signal: it transforms a real signal into a complex signal > with only the positive part of the frequency. > What do we gain? We have been using the double sided spectrum taking > advantage of the complex sinusoids. We could have used simple trigonometric > sines and cosine to get the spectrum instead. The spectrum would have been > positive and one-sided from the beginning. What am i missing? > > thanks for any clarification > fisico30
I have been wondering all the same issues and I would really recommend you to take some time and read this great tutorial by Dr.Lyons: http://www.dspguru.com/info/tutor/QuadSignals.pdf Trust me... It is really good in clarifying Complex Signal Processing. There is also a good tutorial in this category named "Complex signal processing is not - complex" by K.Martin.
On 7 Feb, 04:25, "fisico30" <marcoscipio...@gmail.com> wrote:
> hello forum, > > I might need some help understanding the usefulness of the complex > analytic signal. > > What we measure are real valued signals whose Fourier transform can be > one-sided,
Assuming you use the real-valued sines and cosines as basis functions, then yes. I haven't used, or even seen that basis used, in anything but the very first intro classes on the Fourier Transform.
> or, if we used complex sinusoids, two-sided and symmetric.
Yes.
> Then > the real signal is made of positive and negative frequencies in equal > amount. The negative sinusoids dont really a physical meaning, I guess.
The have a mathematical purpose. Keep in mind that mathematics and physics are not the same. Physics can be described in terms of maths, but not all maths can be interpreted in terms of physics. IN spatial signal processing (i.e. antennas) the positive and negative spatial frequencies represent signals that arrive from different directions.
> Complex analytic signal: it transforms a real signal into a complex signal > with only the positive part of the frequency.
Yes.
> What do we gain? We have been using the double sided spectrum taking > advantage of the complex sinusoids. We could have used simple trigonometric > sines and cosine to get the spectrum instead. The spectrum would have been > positive and one-sided from the beginning. What am i missing?
If you want to represent the spectrum in terms of sines and cosines, you need two spectra: The sine spectrum and the cosine spectrum. Second, if you want to do any manipulations of that kind of spectrum, you need to manipulate both spectra simultaneously and consitently, which makes a mess. Third, not everything interesting is real-valued. One example is spatial processing, where the Fourier Transforms are done stages, where some stages take as input complex-valued data produced by earlier stages. The complex exponential formulation handles complex-valued data as easily as real-valued data. So, the spectrum representation based on the complex exponential is far easier to use in practice, and is far more flexible than the sine & cosine representation. The price one pays is to find a way to mentally handle the negative frequencies when dealing with time signals. Now the Hilbert transform: The Hilbert transform is one DSPtool among many. It is used now and then to convert the time series data to complex form, which is useful in certain demodulation scehemes to find signal envelopes etc. Rune