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Nana113
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Thread moved from the technical forums to the schoolwork forums
so this condition always applies and the integral always equates to 0 when asking for orthogonality?PeroK said:The inner product of two complex things always involves the complex conjugate of one of the things. In this case:$$\langle u, v \rangle = \int_a^b u(x)^*v(x) \ dx$$There are loads of hits on Goggle if you search for "complex inner product".
That's the definition of orthogonality. It's a generalization of the concept of orthogonality for 2D or 3D vectors. Vectors (or functions) are orthogonal if $$\langle u, v \rangle = 0$$In this case that means that two functions are orthogonal if$$\int_a^b u(x)^*v(x) \ dx = 0$$Nana113 said:so this condition always applies and the integral always equates to 0 when asking for orthogonality?
The reason for this being that the inner product is supposed to satisfy ##\langle v,u\rangle = \langle u,v\rangle^*##, resulting in linearity in one argument and anti-linearity in the other.PeroK said:The inner product of two complex things always involves the complex conjugate of one of the things. In this case:$$\langle u, v \rangle = \int_a^b u(x)^*v(x) \ dx$$There are loads of hits on Goggle if you search for "complex inner product".
It should however be noted that an inner product on a function space will often come with an additional weight function. In this case it does not, but it is good to be aware.PeroK said:In this case that means that two functions are orthogonal if$$\int_a^b u(x)^*v(x) \ dx = 0$$
It also, i'd say the main point, makes it positive definite. ##\langle u, u \rangle > 0## for non zero vectors.Orodruin said:Just to add a couple of things.The reason for this being that the inner product is supposed to satisfy ##\langle v,u\rangle = \langle u,v\rangle^*##, resulting in linearity in one argument and anti-linearity in the other.
PeroK said:The inner product of two complex things always involves the complex conjugate of one of the things. In this case:$$\langle u, v \rangle = \int_a^b u(x)^*v(x) \ dx$$There are loads of hits on Goggle if you search for "complex inner product".
Yeah, but that messes up Dirac notation!pasmith said:An inner product on a complex vector space is by definition linear in its first argument and Hermitian (see eg. here), so that [tex]\langle \alpha u, v \rangle = \alpha \langle u, v \rangle[/tex] but [tex]\langle u, \alpha v \rangle = (\langle \alpha v, u \rangle)^{*} = \alpha^{*}\langle u, v \rangle.[/tex] Hence we must have [tex]\langle u, v \rangle = \int_a^b u(x)v^{*}(x)\,dx.[/tex]
Note: This is the typical definition among mathematicians. Among phycisists, the typical convention is that the linearity is in the second argument. One needs to be careful to ensure oneself which convention a particular source uses.pasmith said:An inner product on a complex vector space is by definition linear in its first argument
Because Dirac notation is typically used by physicists. See above.PeroK said:Yeah, but that messes up Dirac notation!
Looks fine to me.Nana113 said:so my working out of the question attached is correct?
Orthogonality of 2 complex exponentials refers to the property where the inner product of two complex exponential functions is equal to zero. Mathematically, this can be represented as ∫(e^(jω1t))(e^(-jω2t)) dt = 0 for ω1 ≠ ω2.
The orthogonality of 2 complex exponentials is fundamental in signal processing as it allows for the decomposition of signals into simpler components. By leveraging this property, signals can be efficiently analyzed and processed using techniques such as Fourier analysis and signal modulation.
Yes, the concept of orthogonality of complex exponentials can be extended to more than 2 functions. In general, a set of complex exponentials is said to be orthogonal if the inner product between any two distinct functions in the set is zero. This property is widely used in various mathematical and engineering applications.
The orthogonality of complex exponentials plays a crucial role in the theory of Fourier transform. In particular, the Fourier transform decomposes a signal into its frequency components by representing the signal as a linear combination of complex exponential functions. The orthogonality property ensures that the frequency components are isolated and can be analyzed independently.
Violating the orthogonality of complex exponentials can lead to signal distortion and interference in signal processing applications. When the orthogonality property is not satisfied, it becomes challenging to accurately analyze and manipulate signals. Therefore, ensuring the orthogonality of complex exponentials is crucial for maintaining the integrity of signal processing systems.