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From: fisico32 on 3 May 2010 17:20 hello forum, Two sinusoids of different frequency w1 and w2 are orthogonal if integral of their product (one time the complex conj of the other) gives zero. That is only true if the integral of integration is T, and and the period of w1 and the period of w2 fit an integer number of times in T... That means that two sinusoids are orthogonal only if their frequencies are different and one an integer multiple of the other.... Correct? In all other cases where the two frequencies are different and not multiple of each other the integral is nonzero....or is there some interval of integration over which they are actually orthogonal? For instance, w1=(4pi) rad/s w2=(3.2pi) rad/s y(t)=sin(w1*t+phi) x(t)=sin(w2*t+theta) Is the integral zero for certain limits of integration? How do I find the limits? Clearly the phases theta and phi matter.... thanks fisico32
From: Dirk Bell on 3 May 2010 17:29 On May 3, 5:20 pm, "fisico32" <marcoscipioni1 (a)n_o_s_p_a_m.gmail.com>wrote: > hello forum, > > Two sinusoids of different frequency w1 and w2 are orthogonal if integral > of their product (one time the complex conj of the other) gives zero. > That is only true if the integral of integration is T, and and the period > of w1 and the period of w2 fit an integer number of times in T... > > That means that two sinusoids are orthogonal only if their frequencies are > different and one an integer multiple of the other.... > Correct? No. Both periods dividing into T does not mean one freq is an integer multiple of the other. Ex. f1=2Hz, f2=3Hz, T=1. > > In all other cases where the two frequencies are different and not multiple > of each other the integral is nonzero....or is there some interval of > integration over which they are actually orthogonal? > > For instance, > w1=(4pi) rad/s > w2=(3.2pi) rad/s > > y(t)=sin(w1*t+phi) > x(t)=sin(w2*t+theta) > Is the integral zero for certain limits of integration? How do I find the > limits? Clearly the phases theta and phi matter.... > > thanks > fisico32
From: fisico32 on 3 May 2010 17:40 >On May 3, 5:20=A0pm, "fisico32" <marcoscipioni1 (a)n_o_s_p_a_m.gmail.com>>wrote: >> hello forum, >> >> Two sinusoids of different frequency w1 and w2 are orthogonal if integral >> of their product (one time the complex conj of the other) gives zero. >> That is only true if the integral of integration is T, and and the period >> of w1 and the period of w2 fit an integer number of times in T... >> >> That means that two sinusoids are orthogonal only if their frequencies ar= >e >> different and one an integer multiple of the other.... >> Correct? > >No. Both periods dividing into T does not mean one freq is an integer >multiple of the other. > >Ex. f1=3D2Hz, f2=3D3Hz, T=3D1. > >> >> In all other cases where the two frequencies are different and not multip= >le >> of each other the integral is nonzero....or is there some interval of >> integration over which they are actually orthogonal? >> >> For instance, >> w1=3D(4pi) rad/s >> w2=3D(3.2pi) rad/s >> >> y(t)=3Dsin(w1*t+phi) >> x(t)=3Dsin(w2*t+theta) >> Is the integral zero for certain limits of integration? How do I find the >> limits? Clearly the phases theta and phi matter.... >> >> thanks >> fisico32 =A0 > Hello Dirk, thanks for the reply. you are right. I guess I meant that if both frequencies have periods T1 and T2 that fit a different integer number or times in the intervalof integration T, then their frequencies w1 and w2 are automatically multiples of each other and the signals are orthogonal over T. My question is about orthogonality between two frequencies that are not multiple of each other. Over which interval are they orthogonal? >
From: Jerry Avins on 3 May 2010 18:15 On 5/3/2010 5:40 PM, fisico32 wrote: >> On May 3, 5:20=A0pm, "fisico32"<marcoscipioni1 (a)n_o_s_p_a_m.gmail.com>>> wrote: >>> hello forum, >>> >>> Two sinusoids of different frequency w1 and w2 are orthogonal if > integral >>> of their product (one time the complex conj of the other) gives zero. >>> That is only true if the integral of integration is T, and and the > period >>> of w1 and the period of w2 fit an integer number of times in T... >>> >>> That means that two sinusoids are orthogonal only if their frequencies > ar= >> e >>> different and one an integer multiple of the other.... >>> Correct? >> >> No. Both periods dividing into T does not mean one freq is an integer >> multiple of the other. >> >> Ex. f1=3D2Hz, f2=3D3Hz, T=3D1. >> >>> >>> In all other cases where the two frequencies are different and not > multip= >> le >>> of each other the integral is nonzero....or is there some interval of >>> integration over which they are actually orthogonal? >>> >>> For instance, >>> w1=3D(4pi) rad/s >>> w2=3D(3.2pi) rad/s >>> >>> y(t)=3Dsin(w1*t+phi) >>> x(t)=3Dsin(w2*t+theta) >>> Is the integral zero for certain limits of integration? How do I find > the >>> limits? Clearly the phases theta and phi matter.... >>> >>> thanks >>> fisico32 =A0 >> > > Hello Dirk, > thanks for the reply. you are right. > I guess I meant that if both frequencies have periods T1 and T2 that fit a > different integer number or times in the intervalof integration T, then > their frequencies w1 and w2 are automatically multiples of each other and > the signals are orthogonal over T. > > My question is about orthogonality between two frequencies that are not > multiple of each other. Over which interval are they orthogonal? Aside from Dirk's correction, still no. A sine and cosine of the same frequency are orthogonal. All sinusoids of different frequencies are orthogonal in that the time-average of their products goes asymptotically to zero as the period of integration becomes progressively longer. The orthogonality is the basis for the Fourier transform. The residual product sum for short integration times gives rise to what we call leakage. Jerry -- "I view the progress of science as ... the slow erosion of the tendency to dichotomize." --Barbara Smuts, U. Mich. �����������������������������������������������������������������������
From: Greg Berchin on 3 May 2010 18:25
On Mon, 03 May 2010 16:40:46 -0500, "fisico32" <marcoscipioni1 (a)n_o_s_p_a_m.gmail.com> wrote:>I guess I meant that if both frequencies have periods T1 and T2 that fit a >different integer number or times in the intervalof integration T, then >their frequencies w1 and w2 are automatically multiples of each other Nope. As Dirk mentioned, if one signal is 2 Hz and the other is 3 Hz, then the interval of integration T is 1 second. But 2 Hz is not a multiple of 3 Hz. They are both multiples of 1 Hz. That's the key: the integration period T is the reciprocal of the greatest common factor of the two frequencies. Example: f1 = 14 Hz, f2 = 16 Hz. 14 = 2 * 7 16 = 2 * 8 The largest common factor is 2. The interval T is 1/2 second. One-half second is 7 periods of f1 and 8 periods of f2. If the two frequencies have an irrational relationship, then T is infinite. Example: f1 = 2 Hz, f2 = pi Hz. There is no common factor. >My question is about orthogonality between two frequencies that are not >multiple of each other. Over which interval are they orthogonal? See above. Greg |