# negative frequency and Hilbert transform

Started by 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:

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
```