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formulas_midterm

# formulas_midterm - For x t = ∑ ∞ k =-∞ δ t-kT we...

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UCSB Fall 2007 ECE 130A: Formulas for Midterm Examination INSTRUCTIONS: The exam is closed book, closed notes, except for the two sides of handwritten notes that you are allowed, and this sheet of formulas. Indicator function: The indicator function of a set A is de±ned as I A ( t ) = b 1 , t A 0 , else Euler’s Identity e = cos θ + j sin θ This implies that cos θ = e + e - 2 , sin θ = e - e - 2 j Convolution: ( x 1 * x 2 )( t ) = i -∞ x 1 ( τ ) x 2 ( t - τ ) (for aperiodic signals x 1 and x 2 ). FOURIER SERIES For a periodic waveform x ( t ) with fundamental period T and fundamental frequency ω 0 , the Fourier series coe²cients { a k } satisfy x ( t ) = s k = -∞ a k e jkω 0 t where a k = 1 T I T x ( t ) e - jkω 0 t dt Fourier Series Example:
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Unformatted text preview: 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 . Signal Fourier Series Coe±cients x ( t ) a k y ( t ) b k Ax ( t ) + By ( t ) Aa k + Bb k x ( t-t ) a k e-jkω t x (-t ) a-k x * ( t ) a *-k x ( t ) y ( t ) ∑ ∞ l =-∞ a l b k-l y ( t ) = dx ( t ) dt b k = jkω a k Parseval’s relation for Fourier series: 1 T i T | x ( t ) | 2 dt = ∑ k | a k | 2...
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