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formulas_final - UCSB Fall 2007 ECE 130A Formulas for Final...

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Unformatted text preview: UCSB Fall 2007 ECE 130A: Formulas for Final Examination INSTRUCTIONS: The exam is closed book, closed notes. In addition to two sides of handwrittend notes, this is the only material you are allowed to consult. Sinc function: sinc( x ) = braceleftBigg sin πx πx x negationslash = 0 1 x = 0 Euler’s Identity e jθ = cos θ + j sin θ This implies that cos θ = e jθ + e − jθ 2 , sin θ = e jθ − e − jθ 2 j Convolution: ( x 1 ∗ x 2 )( t ) = integraltext ∞ −∞ x 1 ( τ ) x 2 ( t − τ ) dτ (for aperiodic signals x 1 and x 2 ). FOURIER SERIES For a periodic waveform x ( t ) with fundamental period T and fundamental frequency ω , the Fourier series coefficients { a k } satisfy x ( t ) = ∞ summationdisplay k = −∞ a k e jkω t where a k = 1 T integraldisplay T x ( t ) e − jkω t dt Fourier Series Example: For x ( t ) = ∑ ∞ k = −∞ δ ( t − kT ), we have a k = 1 T for all k . Fourier Series Properties: In the following table, assume that both x ( t ) and y ( t ) have fundamental period T ....
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formulas_final - UCSB Fall 2007 ECE 130A Formulas for Final...

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