Hi,
The problem to be solved is -> whether the below digital signal is periodic and
if it is, then what is its fundamental period?
x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
But my question is how does one solve such complex trigonometric functions to
reach an answer?
I don't remember anything beyond the below expansions in trigonometry:
(cos(x) - cos(y))
(cos(x) + cos(y))
(sin(x) - sin(y))
(sin(x) + sin(y))
Solving x(n) into one single sinusoid and finding its periodicity is driving me
crazy!
Is there a simpler approach to find the needed periodic frequency other than
going into detailed mathematical calculation. If so how?
I think I have missed big time trigonometry classes!! Please refer me to some
good books/sites as well if you could, to solve these kind of tricky questions
beyond the normal cos and sin matters.
Thanks in advance,
Pallavi
Solving complex trignometric equations
Started by ●January 23, 2009
Reply by ●January 26, 20092009-01-26
Thanks Jonas, now I will quit condensing x(n) equation into a single
sinusoid.
Thanks Prof Ramdas, the solution was well explained and made me understand it in one-shot!
But what if one of these trigonometric component was non-periodic? Will the resulting x(n) be still periodic since the other two components are periodic? If yes, how can the resulting periodic frequency be calculated in that case?
And what rule needs to be applied in the case where y(n) is the result of product of two sinusoids instead of their sum? In the below case, both the components are periodic.
y(n) = cos(pi*n/5) * cos(pi*n/8)
Regards
Pallavi
From: Jonas Svensson
To: g...@yahoo.com
Sent: Friday, January 23, 2009 1:35:17 PM
Subject: Re: [audiodsp] Solving complex trignometric equations
Hi
The function you wrote is peridocid, as it only contains trigonometric function which are periodic. It can not be reduced to a function of a single frequency since different tones (sinuoids) are orthogonal, which is the basis of Fourier analysis. What you have is a function with three frequency components. The fundamental frequency would be 1/16 Hz wih overtones at 1/8 Hz and 1/4 Hz for n=1.
________________________________
From: Ramdas Kumaresan
To: g...@yahoo.com
Sent: Friday, January 23, 2009 1:28:57 PM
Subject: Re: [audiodsp] Solving complex trignometric equations
To check if a discrete time signal x(n) is periodic,
you need to find the smallest value of the integer K,
for which x(n)=x(n+K).
For example, taking the first term on the right side of
your expression for x(n),
check if cos(pi*n/2)=cos(pi(n+K)/2).
For this to be true, pi*K/2 must be an integer multiple of
2*pi, that is it must be of the form 2*pi*N, where N is an integer.
Repeat the same procedure for each term on the right side.
Then pick among those integers N, the smallest value K
for which all three terms are periodic with same period K.
Hope it helps,
Best wishes
Ramdas Kumaresan
Professor of Electrical Engineering
University of Rhode Island
Kingston, RI USA
Thanks Prof Ramdas, the solution was well explained and made me understand it in one-shot!
But what if one of these trigonometric component was non-periodic? Will the resulting x(n) be still periodic since the other two components are periodic? If yes, how can the resulting periodic frequency be calculated in that case?
And what rule needs to be applied in the case where y(n) is the result of product of two sinusoids instead of their sum? In the below case, both the components are periodic.
y(n) = cos(pi*n/5) * cos(pi*n/8)
Regards
Pallavi
From: Jonas Svensson
To: g...@yahoo.com
Sent: Friday, January 23, 2009 1:35:17 PM
Subject: Re: [audiodsp] Solving complex trignometric equations
Hi
The function you wrote is peridocid, as it only contains trigonometric function which are periodic. It can not be reduced to a function of a single frequency since different tones (sinuoids) are orthogonal, which is the basis of Fourier analysis. What you have is a function with three frequency components. The fundamental frequency would be 1/16 Hz wih overtones at 1/8 Hz and 1/4 Hz for n=1.
________________________________
From: Ramdas Kumaresan
To: g...@yahoo.com
Sent: Friday, January 23, 2009 1:28:57 PM
Subject: Re: [audiodsp] Solving complex trignometric equations
To check if a discrete time signal x(n) is periodic,
you need to find the smallest value of the integer K,
for which x(n)=x(n+K).
For example, taking the first term on the right side of
your expression for x(n),
check if cos(pi*n/2)=cos(pi(n+K)/2).
For this to be true, pi*K/2 must be an integer multiple of
2*pi, that is it must be of the form 2*pi*N, where N is an integer.
Repeat the same procedure for each term on the right side.
Then pick among those integers N, the smallest value K
for which all three terms are periodic with same period K.
Hope it helps,
Best wishes
Ramdas Kumaresan
Professor of Electrical Engineering
University of Rhode Island
Kingston, RI USA
Reply by ●January 26, 20092009-01-26
G Pallavi-
> The problem to be solved is -> whether the below digital signal is periodic and if it is, then what is its fundamental period?
>
> x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
The composite waveform is periodic for the "slowest" term, or in this case every 16
samples due to the second term. Note there is an initial 'negative' offset due to
the pi/3 constant in the third term.
-Jeff
> But my question is how does one solve such complex trigonometric functions to reach an answer?
>
> I don't remember anything beyond the below expansions in trigonometry:
> (cos(x) - cos(y))
> (cos(x) + cos(y))
> (sin(x) - sin(y))
> (sin(x) + sin(y))
>
> Solving x(n) into one single sinusoid and finding its periodicity is driving me crazy!
> Is there a simpler approach to find the needed periodic frequency other than going into detailed mathematical calculation. If so how?
> I think I have missed big time trigonometry classes!! Please refer me to some good books/sites as well if you could, to solve these kind of tricky questions beyond the normal cos and sin matters.
>
> Thanks in advance,
> Pallavi
> The problem to be solved is -> whether the below digital signal is periodic and if it is, then what is its fundamental period?
>
> x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
The composite waveform is periodic for the "slowest" term, or in this case every 16
samples due to the second term. Note there is an initial 'negative' offset due to
the pi/3 constant in the third term.
-Jeff
> But my question is how does one solve such complex trigonometric functions to reach an answer?
>
> I don't remember anything beyond the below expansions in trigonometry:
> (cos(x) - cos(y))
> (cos(x) + cos(y))
> (sin(x) - sin(y))
> (sin(x) + sin(y))
>
> Solving x(n) into one single sinusoid and finding its periodicity is driving me crazy!
> Is there a simpler approach to find the needed periodic frequency other than going into detailed mathematical calculation. If so how?
> I think I have missed big time trigonometry classes!! Please refer me to some good books/sites as well if you could, to solve these kind of tricky questions beyond the normal cos and sin matters.
>
> Thanks in advance,
> Pallavi
Reply by ●January 26, 20092009-01-26
hello,
I remember reading similar problems in "Signals and Systems" - Simon Haykin
I dont recall perfectly but the approach is the following
x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
1. find the periods of individual sinusoids.
2. make sure the individual periods are rational w.r.t each other
3. LCM ( or HCF not sure) of the individual periods gives u the period of
the function x(n).
.
On Fri, Jan 23, 2009 at 3:31 AM, wrote:
> Hi,
>
> The problem to be solved is -> whether the below digital signal is periodic
> and if it is, then what is its fundamental period?
>
> x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
>
> But my question is how does one solve such complex trigonometric functions
> to reach an answer?
>
> I don't remember anything beyond the below expansions in trigonometry:
> (cos(x) - cos(y))
> (cos(x) + cos(y))
> (sin(x) - sin(y))
> (sin(x) + sin(y))
>
> Solving x(n) into one single sinusoid and finding its periodicity is
> driving me crazy!
> Is there a simpler approach to find the needed periodic frequency other
> than going into detailed mathematical calculation. If so how?
> I think I have missed big time trigonometry classes!! Please refer me to
> some good books/sites as well if you could, to solve these kind of tricky
> questions beyond the normal cos and sin matters.
>
> Thanks in advance,
> Pallavi
>
>
--
To accomplish great things, we must dream as well as act
I remember reading similar problems in "Signals and Systems" - Simon Haykin
I dont recall perfectly but the approach is the following
x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
1. find the periods of individual sinusoids.
2. make sure the individual periods are rational w.r.t each other
3. LCM ( or HCF not sure) of the individual periods gives u the period of
the function x(n).
.
On Fri, Jan 23, 2009 at 3:31 AM, wrote:
> Hi,
>
> The problem to be solved is -> whether the below digital signal is periodic
> and if it is, then what is its fundamental period?
>
> x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
>
> But my question is how does one solve such complex trigonometric functions
> to reach an answer?
>
> I don't remember anything beyond the below expansions in trigonometry:
> (cos(x) - cos(y))
> (cos(x) + cos(y))
> (sin(x) - sin(y))
> (sin(x) + sin(y))
>
> Solving x(n) into one single sinusoid and finding its periodicity is
> driving me crazy!
> Is there a simpler approach to find the needed periodic frequency other
> than going into detailed mathematical calculation. If so how?
> I think I have missed big time trigonometry classes!! Please refer me to
> some good books/sites as well if you could, to solve these kind of tricky
> questions beyond the normal cos and sin matters.
>
> Thanks in advance,
> Pallavi
>
>
--
To accomplish great things, we must dream as well as act
Reply by ●January 26, 20092009-01-26
From sin(pi*n/8), I come at a conclusion that period is 16.
2009-01-23 17:31:34=A...@yahoo.com ะด
Hi,
The problem to be solved is -> whether the below digital signal is periodic and if it is, then what is its fundamental period?
x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
But my question is how does one solve such complex trigonometric functions to reach an answer?
I don't remember anything beyond the below expansions in trigonometry:
(cos(x) - cos(y))
(cos(x) + cos(y))
(sin(x) - sin(y))
(sin(x) + sin(y))
Solving x(n) into one single sinusoid and finding its periodicity is driving me crazy!
Is there a simpler approach to find the needed periodic frequency other than going into detailed mathematical calculation. If so how?
I think I have missed big time trigonometry classes!! Please refer me to some good books/sites as well if you could, to solve these kind of tricky questions beyond the normal cos and sin matters.
Thanks in advance,
Pallavi
_____________________________________
2009-01-23 17:31:34=A...@yahoo.com ะด
Hi,
The problem to be solved is -> whether the below digital signal is periodic and if it is, then what is its fundamental period?
x(n) = cos(pi*n/2) - sin(pi*n/8) + 3cos(pi*n/4 + pi/3)
But my question is how does one solve such complex trigonometric functions to reach an answer?
I don't remember anything beyond the below expansions in trigonometry:
(cos(x) - cos(y))
(cos(x) + cos(y))
(sin(x) - sin(y))
(sin(x) + sin(y))
Solving x(n) into one single sinusoid and finding its periodicity is driving me crazy!
Is there a simpler approach to find the needed periodic frequency other than going into detailed mathematical calculation. If so how?
I think I have missed big time trigonometry classes!! Please refer me to some good books/sites as well if you could, to solve these kind of tricky questions beyond the normal cos and sin matters.
Thanks in advance,
Pallavi
_____________________________________
