EECS
final-exam

# final-exam - EECS 20n Structure and Interpretation of...

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Complex exponential Fourier series synthesis and analysis equations for a periodic discrete-time signal having period p : x ( n ) = X k = h p i X k e ikω 0 n ←→ X k = 1 p X n = h p i x ( n ) e - ikω 0 n , where p = 2 π ω 0 and h p i denotes a suitable contiguous discrete interval of length p (for example, X k = h p i can denote p - 1 X k =0 ). Complex exponential Fourier series synthesis and analysis equations for a periodic continuous-time signal having period p : x ( t ) = X k = -∞ X k e ikω 0 t ←→ X k = 1 p Z h p i x ( t ) e - ikω 0 t dt , where p = 2 π ω 0 and h p i denotes a suitable continuous interval of length p (for example, Z h p i can denote Z p 0 ). Discrete-time Fourier transform (DTFT) synthesis and analysis equations for a discrete-time signal: x ( n ) = 1 2 π Z h 2 π i X ( ω ) e iωn ←→ X ( ω ) = X n = -∞ x ( n ) e - iωn , where h 2 π i denotes a suitable continuous interval of length 2 π (for example, Z h 2 π i can denote Z 2 π 0 or Z π - π ). Continuous-time Fourier transform (CTFT) synthesis and analysis equations for a continuous-time signal: x ( t ) = 1 2 π Z -∞ X ( ω ) e iωt ←→ X ( ω ) = Z -∞ x ( t ) e - iωt dt . 2
Problem 1 (5 Points Total) A causal system is initially at rest (i.e., all initial con- ditions are zero) and is described by the following delay-adder-gain block diagram. The block D denotes a delay by one sample; that is, if the input to the delay block D is a signal p , the output of the delay block is q , where q ( n ) = p ( n - 1) for all n .

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