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Suppose we want to transmit data over a period of seconds using a real-valued signal, denoted by [tex]f(t)[/tex]. The rectangular function [tex]rect(t)[/tex] is defined over this period.

We have a set of orthonormal basis functions [tex]\{\phi_n(t)\}[/tex] with which we wish to approximate [tex]f(t)[/tex] such that the energy in the error signal is minimized.

Calculate the optimum coefficients [tex]c_n[/tex] that minimize the energy of the error signal.

Answer :

The question concerns calculating the optimum coefficients for representing a real-valued signal using orthonormal basis functions to minimize error, which relates to Fourier series and transforms in signal processing.

Fourier Series and Transforms in Signal Processing

The student's question pertains to the field of signal processing, an area within electrical engineering which involves the analysis and manipulation of signals. In signal processing, a common task is to represent a signal using a set of orthonormal basis functions, such as sine and cosine waves, which can fully describe a function's behavior over time or space. Fourier series are used for periodic signals, while Fourier transforms are used for non-periodic signals. To minimize the error between an original signal and its approximation, the optimum coefficients would be determined by projecting the original signal onto the basis functions, resulting in the least-squares approximation of the original signal.To calculate the optimum coefficients, one must perform an inner product between the signal and each basis function. In the case of a Fourier series, these coefficients are complex numbers reflecting the amplitude and phase of each frequency component in the series. Essentially, applying Fourier's theorem and the Fourier inversion theorem, allows any well-behaved signal to be represented as a sum of sines and cosines (or an integral), minimizing the residual energy not captured by the approximation.

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