Signal multiplication with complex number

Hello,

I would like to know the implication of multiplying a discrete sinusoidal
signal with a complex number. A simple example in Matlab would be very
helpful.

Thanks
---------------------------------------
Posted through http://www.DSPRelated.com

On Sat, 30 Jan 2016 12:28:14 -0600, Sharan123 wrote:

> Hello,
> 
> I would like to know the implication of multiplying a discrete
> sinusoidal signal with a complex number. A simple example in Matlab
> would be very helpful.

The question is too general.  You end up with a complex-valued discrete 
sinusoid.  Not very interesting.

What are you trying to _do_?

-- 
www.wescottdesign.com

On Sat, 30 Jan 2016 12:28:14 -0600, "Sharan123" <99077@DSPRelated>
wrote:

>Hello,
>
>I would like to know the implication of multiplying a discrete sinusoidal
>signal with a complex number. A simple example in Matlab would be very
>helpful.
>
>Thanks
>---------------------------------------
>Posted through http://www.DSPRelated.com

What do you mean by "implication"?   The result is complex-valued, if
that's what you're after.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

Well you see, sometimes a discrete sinusoid meets a complex number, and they decide to multiply (this is now legal in Cambridge Massachusetts where I work). Their child is both discrete and complex (yes I am aware that discrete and discreet are not the same). 

Bob

On Sat, 30 Jan 2016 18:36:26 -0800 (PST), radams2000@gmail.com wrote:

>Well you see, sometimes a discrete sinusoid meets a complex number, and the=
>y decide to multiply (this is now legal in Cambridge Massachusetts where I =
>work). Their child is both discrete and complex (yes I am aware that discre=
>te and discreet are not the same).=20
>
>Bob

Is the interaction of indiscreet sinusoids and complex numbers still
illegal, then?

Reminds me a little of that old, elegant prose, Impure Mathematics
that used to circulate regularly decades ago:

(archived here among other places:
https://www-users.cs.york.ac.uk/susan/joke/polly.htm )


Once upon a time (1/t), pretty little Polly Nomial was strolling
across a field of vectors when she came to the edge of a singularly
large matrix.

Now Polly was convergent and her mother had made it an absolute
condition that she must never enter such an array without her brackets
on. Polly, however, who had changed her variables that morning and was
feeling particularly badly behaved, ignored this condition on the
grounds that it was insufficient and made her way in amongst the
complex elements.

Rows and columns enveloped her on all sides. Tangents approached her
surface. She became tensor and tensor. Quite suddenly, three branches
of a hyperbola touched her at a single point. She oscillated
violently, lost all sense of directrix and went completely divergent.
As she reached a turning point she tripped over a square root which
was protruding from the erf and plunged headlong down a steep
gradient. When she was differentiated once more she found herself,
apparently alone, in a non-euclidean space.

She was being watched, however. That smooth operator, Curly Pi, was
lurking inner product. As his eyes devoured her curvilinear
coordinates, a singular expression crossed his face. Was she still
convergent, he wondered. He decided to integrate improperly at once.

Hearing a vulgar function behind her, Polly turned round and saw Curly
Pi approaching with his power series extrapolated. She could see at
once, by his degenerate conic and his dissipative terms, that he was
bent on no good.

"Eureka" she gasped.

"Ho, ho," he said. "What a symmetric little Polynomial you are. I can
see you're bubbling over with secs".

"O Sir," she protested, "keep away from me. I haven't got my brackets
on."

"Calm yourself, my dear," said our suave operator, "your fears are
purely imaginary "

"i, i," she thought, "perhaps he's homogenous then?".

"What order are you," the brute demanded.

"Seventeen," replied Polly.

Curly leered. "I suppose you've never been operated on yet?" he asked.

"Of course not", Polly cried indignantly. "I'm absolutely convergent."

"Come, come," said Curly. "Let's off to a decimal place I know and
I'll take you to the limit."

"Never," gasped Polly.

"Exchlf," he swore, using the vilest oath he knew. His patience was
gone. Coshing her over the coefficient with a log until she was
powerless, Curly removed her discontinuities. He stared at her
significant places and began to smooth her points of inflexion. Poor
Polly. All was up. She felt his hand tending to her asymptotic limit.
Her convergence would soon be gone forever.

There was no mercy, for Curly was a heavyside operator. He integrated
by parts. He integrated by partial fractions. The complex beast even
went all the way around and did a contour integration. What an
indignity. To be multiply connected on her first integration. Curly
went on operating until he was absolutely and completely orthogonal.

When Polly got home that evening, her mother noticed that she had been
truncated in several places. But it was too late to differentiate now.
As the months went by, Polly increased monotonically. Finally she
generated a small but pathological function which left surds all over
the place until she was driven to distraction.

The moral of this sad story is this: If you want to keep your
expressions convergent, never allow them a single degree of freedom.



Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On Sat, 30 Jan 2016 12:28:14 -0600, Sharan123 wrote:

> I would like to know the implication of multiplying a discrete sinusoidal
> signal with a complex number.

Sinusoid or complex exponential? Signal processing theory tends to use the
latter:

	e^(i*x) = cos(x) + i*sin(x)

The main advantage of this over a sinusoid is that you can perform a phase
shift by multiplying by a complex constant.

Hello all,

Some context behind my question - this is related to receiver stages in
wireless communication. In many algorithms related to multiple receiver
antennae there is a reference to channel gain, which is complex valued. It
talks about multiplying received signal with complex channel gain
factors.

I am trying to understand what happens when this is done. I guess it
alters both amplitude and phase of the signal. Also, after the
multiplication, how is the signal converted back into real domain from
complex domain?

PS: I did not want to go into too much of background, as I was trying to
understand fundamentally what happens if a signal is multiplied by a
complex number


---------------------------------------
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On 1.2.16 17:10, Sharan123 wrote:
> Hello all,
>
> Some context behind my question - this is related to receiver stages in
> wireless communication. In many algorithms related to multiple receiver
> antennae there is a reference to channel gain, which is complex valued. It
> talks about multiplying received signal with complex channel gain
> factors.
>
> I am trying to understand what happens when this is done. I guess it
> alters both amplitude and phase of the signal. Also, after the
> multiplication, how is the signal converted back into real domain from
> complex domain?
>
> PS: I did not want to go into too much of background, as I was trying to
> understand fundamentally what happens if a signal is multiplied by a
> complex number


This means simply that the channel can change both the
amplitude and the phase of the signal.

-- 

-TV

Sharan123 <99077@DSPRelated> wrote:

>Some context behind my question - this is related to receiver stages in
>wireless communication. In many algorithms related to multiple receiver
>antennae there is a reference to channel gain, which is complex valued. It
>talks about multiplying received signal with complex channel gain
>factors.

>I am trying to understand what happens when this is done. I guess it
>alters both amplitude and phase of the signal. Also, after the
>multiplication, how is the signal converted back into real domain from
>complex domain?
 
This stems from the Turin model, which is the basic multipath channel 
model from which other more elaborate channel models are derived.
 
Turin (who passed away a few years back; some of us here will
remember him from UC Berkeley and Teknekron) posited that for a high 
enough carrier frequncy, each reflection would arrive at the antenna 
with a uniform random phase shift on the interval [0, 2*pi), and thus 
the amplitude of this reflection can be represented by a complex
coefficient.  

And further, using this, reflections could then be approximated 
by applying time delays plus complex multiplies by these coefficients
to the complex baseband signal from the transmitter, with the result
forming the complex baseband signal in the receiver.

So the idea is, not only do you model the RF channel in a
simple manner, you conveniently bypass needing to fully model the
upconversion/downconversoin to/from RF.  Of course you need
to add back in impairments that are bypassed (important ones
being PA non-linearity, receiver noise figure, local oscillator
phase noise, frequency offset and doppler).

So to answer your question in the last sentence above: you don't.
You model the whole thing as a complex signal path, and never convert
to or from a real signal.

Hope this helps.

Steve

On Mon, 01 Feb 2016 09:10:42 -0600, "Sharan123" <99077@DSPRelated>
wrote:

>Hello all,
>
>Some context behind my question - this is related to receiver stages in
>wireless communication. In many algorithms related to multiple receiver
>antennae there is a reference to channel gain, which is complex valued. It
>talks about multiplying received signal with complex channel gain
>factors.
>
>I am trying to understand what happens when this is done. I guess it
>alters both amplitude and phase of the signal. Also, after the
>multiplication, how is the signal converted back into real domain from
>complex domain?
>
>PS: I did not want to go into too much of background, as I was trying to
>understand fundamentally what happens if a signal is multiplied by a
>complex number

This is generally modelled at baseband while the signal also has
complex-valued samples.     It is possible to do it at an IF while the
signal is real-valued, but it is far simpler to do at baseband with
complex-valued signal samples.

So a complex-valued channel gain per antenna gets multiplied by the
complex-valued baseband signal received at that antenna.    From this
perspective the phase is really only relevant to model the phase
differences between antennas.   For a single antenna system a
flat-fading channel would only require a scalar amplitude coefficient,
but for multiple antennas the relative phases matter.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com