The Idea of Orthogonality
Explore why many DSP algorithms rely on orthogonality.
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To understand the idea of orthogonality, we consider both real and complex signals.
Real sinusoids
The figure below shows the continuous-time versions of two sinusoids containing samples, one with frequency and the other with frequency . The discrete frequencies are and , but it’s easier to understand this idea with a continuous curve than with a discrete sinusoid.
Observe the “Q” part (the vertical plane at the back of the screen) of the two waves and look at the number of samples in each.
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The wave with frequency has one cycle in samples.
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The wave with frequency has one cycle in samples.
Clearly, the wave with frequency completes one cycle for the positive half of the first sinusoid and another cycle for the negative half of the first sinusoid. Due to these opposite signs, a summation of their sample-by-sample product is zero. Orthogonality between two signals means that a summation of their sample-by-sample product (i.e., correlation at time ) is zero.
Mathematically, we write for the “Q” part in the figure above:
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