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Exam 2 cheat sheet

Exam 2 cheat sheet - In general every time sequence x[n can...

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In general, every time sequence x [ n ] can be expressed as = + - + - +…= = ∞ - xn x0δn x1δn 1 x2δn 2 k 0 xkδn k If the system is linear and time invariant (LTI), then we have δ [ n ] → h [ n ] (impulse response) δ [ n k ] → h [ n k ] (shifting) x [ k ] δ [ n k ] → x [ k ] h [ n k ] (homogeneity) = ∞ - → = ∞ - k 0 xkδn k k 0 xkhn k (additivity) Discrete-time convolution equation : = = [ - ] yn k 0 xkh n k A system is said to be causal if its current output depends only on its current and past inputs. If the output depends on future inputs, the system is said to be noncausal . A system is causal if and only if h [ n ] = 0 for n < 0 . A digital system is called a finite-impulse-response (FIR) system or filter if its impulse response has a finite number of non-zero entries. A causal FIR filter is said to have length N if h [ N – 1] ≠ 0 and h [ n ] = 0 for all n > N – 1 . If N = 1 , or h [0] = a ≠ 0 and h [ n ] = 0 for n ≠ 0 , then this reduces to y [ n ] = ax [ n ] where a is a constant. (Multiplier). A digital system is called an infinite-impulse-response (IIR) filter if its impulse response has infinitely many non-zero values. Discrete-time difference equation (Delayed form): + - +…+ + - = + - +…+ + - a1yn a2yn 1 aN 1yn N b1xn b2xn 1 bM 1xn M where a i and b i are real constants, and M and N are positive integers. With a 1 ≠ 0 is the most general form of difference equations to describe LTIL (lumped LTI) and causal systems. If a N +1 ≠ 0 and b M +1 ≠ 0 , the difference equation is said to have order max( M, N ) . Nyquist Frequency Range: (- π / T , π / T ] Non-recursive difference equation : = + - +…+ + [ - ] yn b1xn b2xn 1 bM 1x n M Recursive difference equation : =- - -…- + - + + - +…+ + [ - ] yn a2yn 1 aN 1yn N b1xn b2xn 1 bM 1x n M

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Exam 2 cheat sheet - In general every time sequence x[n can...

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