Is it possible to reproduce a small bandwidth (~10 hz) audio signal using a sine wave of variable frequency and amplitude?

I added 3 sine waves together to produce a signal.  

Sine Wave 1: Amplitude=1, Frequency=55 hz
Sine Wave 2: Amplitude=0.39, Frequency=52.87 hz
Sine Wave 3: Amplitude=0.34, Frequency=57.54 hz

The starting phase for all 3 sine waves is zero.

Is it possible for me to reproduce this signal using a single sine wave of
variable frequency and amplitude?  (i mean use a sine oscillator, and
control the frequency using one data table and control the amplitude using
another data table)

If so, what method would you suggest for deriving the frequency control
data?


I wrote a program in Scilab to derive the amplitude curve, it works
beautifully.  


I started on a program in Scilab to derive the frequency curve.  The
method that I have in mind is to calculate a frequency value for each
local maxima in the signal by calculating the time difference between each
local maxima, then interpolate the frequency curve linearly between these
points.  

However, I wonder if there's a better method for deriving the frequency
curve from the signal.  Perhaps some kind of sine wave pattern matching
algorithm.  Can anyone suggest anything?

Thanks, any help would be much appreciated

On 28 Nov, 05:52, "maxplanck" <erik.bo...@comcast.net> wrote:
> I added 3 sine waves together to produce a signal.  
>
> Sine Wave 1: Amplitude=1, Frequency=55 hz
> Sine Wave 2: Amplitude=0.39, Frequency=52.87 hz
> Sine Wave 3: Amplitude=0.34, Frequency=57.54 hz
>
> The starting phase for all 3 sine waves is zero.
>
> Is it possible for me to reproduce this signal using a single sine wave of
> variable frequency and amplitude?  (i mean use a sine oscillator, and
> control the frequency using one data table and control the amplitude using
> another data table)

Maybe what you want is possible, maybe not; it would certainly
become very complicated. The usual theory deals with signals
where *either* frequency *or* amplitude varies. These are called
FM and AM signals, respectively. Analyzing a signal where
both frequency and amplitude varies independently of each other
will be no simple task.

Rune

Rune Allnor wrote:
> On 28 Nov, 05:52, "maxplanck" <erik.bo...@comcast.net> wrote:

>>I added 3 sine waves together to produce a signal.  

>>Sine Wave 1: Amplitude=1, Frequency=55 hz
>>Sine Wave 2: Amplitude=0.39, Frequency=52.87 hz
>>Sine Wave 3: Amplitude=0.34, Frequency=57.54 hz

>>The starting phase for all 3 sine waves is zero.

>>Is it possible for me to reproduce this signal using a single sine wave of
>>variable frequency and amplitude?  (i mean use a sine oscillator, and
>>control the frequency using one data table and control the amplitude using
>>another data table)

I would say that you could always do that, unless you put some
restrictions on the modulations.  You could set the frequency
to zero and modulate the amplitude according to the sum of
three sine waves, for one example.  Otherwise, you have to carefully
change the frequency so that zero crossings come in the right place.
It shouldn't be so hard, though.

> Maybe what you want is possible, maybe not; it would certainly
> become very complicated. The usual theory deals with signals
> where *either* frequency *or* amplitude varies. These are called
> FM and AM signals, respectively. Analyzing a signal where
> both frequency and amplitude varies independently of each other
> will be no simple task.

-- glen

>> Is it possible for me to reproduce this signal using a single sine wave
of variable frequency and amplitude? (

Yes, it is possible, and it is done in many modern cellphone
transmitters.
The idea is to write a signal in so-called "complex baseband
representation".
Basically you multiply with sin(2 pi f t) and cos(2 pi f t) and lowpass
filter according to the bandwidth. f is the frequency of your carrier
signal.

The resulting signal has in-phase and quadrature component, which can be
written as amplitude and phase. Differentiate the phase and you get the
frequency.

The math is in basic communications text books. The example transmitter
architecture is called "polar" architecture, for example used for GSM
EDGE.

It doesn't matter if you have one, two or three sines, what matters is the
rrelationship between your total bandwidth and the mentioned lowpass
filter.

-mn

Thanks for replying.  I started reading about complex baseband
representation, looks like a lot of stuff for me to learn.

Do you know of any software, matlab or scilab signal processing toolboxes,
etc. that can take a signal as an input, and output the amplitude and
frequency or phase data for a given carrier frequency in order to
reproduce the input signal?  

If I read and spend a lot of time researching complex baseband
representation I can probably figure this out on my own, but it would help
me very much to simply be able to use software to get the data I need
without my having to study and understand the methods.


Thanks again for replying

>>> Is it possible for me to reproduce this signal using a single sine
wave
>of variable frequency and amplitude? (
>
>Yes, it is possible, and it is done in many modern cellphone
>transmitters.
>The idea is to write a signal in so-called "complex baseband
>representation".
>Basically you multiply with sin(2 pi f t) and cos(2 pi f t) and lowpass
>filter according to the bandwidth. f is the frequency of your carrier
>signal.
>
>The resulting signal has in-phase and quadrature component, which can be
>written as amplitude and phase. Differentiate the phase and you get the
>frequency.
>
>The math is in basic communications text books. The example transmitter
>architecture is called "polar" architecture, for example used for GSM
>EDGE.
>
>It doesn't matter if you have one, two or three sines, what matters is
the
>rrelationship between your total bandwidth and the mentioned lowpass
>filter.
>
>-mn
>

multiply with exp(-i 2 pi f t) and use lowpass filters on real and
imaginary part. Or a complex-valued filter if you like.
That is your complex baseband signal. Then take magnitude and angle -
voila.

The exp(...) does BTW a downwards shift in frequency by f.

Is there a way to represent a signal using an amplitude and a frequency
component, instead of an amplitude and a phase component?

>multiply with exp(-i 2 pi f t) and use lowpass filters on real and
>imaginary part. Or a complex-valued filter if you like.
>That is your complex baseband signal. Then take magnitude and angle -
>voila.
>
>The exp(...) does BTW a downwards shift in frequency by f.
>
>

maxplanck wrote:
> Is there a way to represent a signal using an amplitude and a frequency
> component, instead of an amplitude and a phase component?

Bad assumption. A sinusoid is characterized by frequency, phase, and 
amplitude, or another set of three parameters. Two are not enough.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

>maxplanck wrote:
>> Is there a way to represent a signal using an amplitude and a
frequency
>> component, instead of an amplitude and a phase component?
>
>Bad assumption. A sinusoid is characterized by frequency, phase, and 
>amplitude, or another set of three parameters. Two are not enough.
>
>Jerry
>-- 
>Engineering is the art of making what you want from things you can get.
>�����������������������������������������������������������������������
>

Sorry, what I meant was: 

Is there a way to represent a signal using a variable amplitude component,
a variable frequency component, and a constant phase component?

From reading some of this text, it seems that the Baseband model of
Bandpass signals represents sinusoids using a variable amplitude component
and a variable phase component, and a constant carrier frequency.  Is this
correct?

http://cmclab.rice.edu/433/notes/Fitz_BandpassNotes.pdf

On Nov 30, 10:09 am, "maxplanck" <erik.bo...@comcast.net> wrote:
> >maxplanck wrote:
> >> Is there a way to represent a signal using an amplitude and a
> frequency
> >> component, instead of an amplitude and a phase component?
>
> >Bad assumption. A sinusoid is characterized by frequency, phase, and
> >amplitude, or another set of three parameters. Two are not enough.
...
> Sorry, what I meant was:
>
> Is there a way to represent a signal using a variable amplitude component,
> a variable frequency component, and a constant phase component?

If the phase is constant, you have DC, not a variable
sinusoid (except for degenerate mathematical models).

Frequency and delta-phase-versus-time are not independent
for frequency modulated sinusoids. Frequency is the rate
of change of the unwrapped phase.  But to specify your
waveform uniquely, an absolute phase reference for at
least one point will be needed .