The continuous Fourier series spans the space of all piecewise continuous, bounded periodic functions. The discrete...

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The continuous Fourier series spans the space of all piecewise continuous, bounded periodic functions. The discrete Fourier series (sometimes erroneously called the discrete Fourier transform) is similar and can be thought of as a change in basis from a time-domain vector to a frequency-domain vector. The following is the general form of the discrete Fourier series synthesis equation: N-1 f(x) = a₁ + [lan sin(nknx) + bn cos(nknx)] n=0 Questions (answer on paper and upload scan): a) Explain why both the sine and cosine are needed to span the infinite space of all periodic functions. b) Use the inner product to show that sine and cosine are orthogonal functions. c) What is an equivalent way to represent a Fourier series using only a single term instead of two terms (sine and cosine)? The continuous Fourier series spans the space of all piecewise continuous, bounded periodic functions. The discrete Fourier series (sometimes erroneously called the discrete Fourier transform) is similar and can be thought of as a change in basis from a time-domain vector to a frequency-domain vector. The following is the general form of the discrete Fourier series synthesis equation: N-1 f(x) = a₁ + [lan sin(nknx) + bn cos(nknx)] n=0 Questions (answer on paper and upload scan): a) Explain why both the sine and cosine are needed to span the infinite space of all periodic functions. b) Use the inner product to show that sine and cosine are orthogonal functions. c) What is an equivalent way to represent a Fourier series using only a single term instead of two terms (sine and cosine)? The continuous Fourier series spans the space of all piecewise continuous, bounded periodic functions. The discrete Fourier series (sometimes erroneously called the discrete Fourier transform) is similar and can be thought of as a change in basis from a time-domain vector to a frequency-domain vector. The following is the general form of the discrete Fourier series synthesis equation: N-1 f(x) = a₁ + [lan sin(nknx) + bn cos(nknx)] n=0 Questions (answer on paper and upload scan): a) Explain why both the sine and cosine are needed to span the infinite space of all periodic functions. b) Use the inner product to show that sine and cosine are orthogonal functions. c) What is an equivalent way to represent a Fourier series using only a single term instead of two terms (sine and cosine)? The continuous Fourier series spans the space of all piecewise continuous, bounded periodic functions. The discrete Fourier series (sometimes erroneously called the discrete Fourier transform) is similar and can be thought of as a change in basis from a time-domain vector to a frequency-domain vector. The following is the general form of the discrete Fourier series synthesis equation: N-1 f(x) = a₁ + [lan sin(nknx) + bn cos(nknx)] n=0 Questions (answer on paper and upload scan): a) Explain why both the sine and cosine are needed to span the infinite space of all periodic functions. b) Use the inner product to show that sine and cosine are orthogonal functions. c) What is an equivalent way to represent a Fourier series using only a single term instead of two terms (sine and cosine)?

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a Explanation for the Need for Both Sine and Cosine To represent a wide range of periodic functions both sine and cosine waves are necessary This is b... View the full answer