Properties of Z-transform Transform 1 Linearity a

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1 Midterm 3 (Fall 6 of EEG:. Thi midterm conit of eight ingle-ided page. The firt three page contain variou table followed by FOUR eam quetion and one etra workheet. You can tear out any page but make ure all the page you turn in have your name and tudent ID on them.. You have one hour to finih thi eam. 3. You are allowed to ue two double-ided page of cheat heet. Good luck! Propertie of Z-tranform Tranform Linearity a ( nt a( nt a X( z a X ( z n Multiply by a a n (nt z X a 3 Time Delay ( nt mt nt mt, m > z m X (z Multiply by n n (nt d z X (z dz 5 Initial Value Theorem ( lim z X ( z 6 Final Value Theorem ( lim ( z X ( 7 Time Convolution m z z ( mt y( nt mt X ( z Y ( z

3 Propertie Time Domain Laplace Tranform a( t a ( t a X( a X ( Linearity K a ( t K a X ( n n n n αt Frequency Shifting e (t F ( α 3 Time Delay ( t a t a e X ( Time Scaling ( α t X α α 5 Time Differentiation d (t X ( ( dt 6 Time Integration t X ( ( τ dτ ( τ dτ 7 Initial Value Theorem lim ( t lim ( ( t X 8 Final Value Theorem lim t ( t lim X ( ( 9 Time Convolution ( t * y( t X ( Y (

4 . (5 point Dicrete Fourier Tranform and Convolution Let (n and y(n be two three-point equence: for n ( n for n y( n for n for n for n for n a. (9 point Compute the 3-point circular convolution z(n between (n and y(n. z( ( 3 z( ( z( ( b. ( point Compute the 5-point DFT X(k for (n. You do not need to implify your anwer. X ( π π X ( ep j ep j 5 5 π 8π X ( ep j ep j 5 5 6π π X (3 ep j ep j 5 5 8π 6π X ( ep j ep j 5 5 c. (6 point In addition to X(k, uppoe you have alo computed the 5-point DFT Y(k for y(n. Without carrying out any computation, comment on whether applying a 5-point DFT on z(n (part a will reult in the product of X(k and Y(k. Eplain your anwer. No. When multiplying X(k with Y(k, one obtain the 5-point DFT of the linear convolution between (n and y(n, not the 3-point circular convolution z(n.

5 . (5 point Fat Fourier Tranform a. (5 point Name one reaon why FFT i very commonly ued in ignal proceing application. FFT can be eecuted very efficiently. b. (5 point We dicu two different FFT implementation in lecture and in homework. Name thee two implementation. Decimation-in-time and Decimation-in-frequency. c. (5 point What i the relationhip between DFT and DTFT? DFT (Dicrete Fourier Tranform are frequency ample of DTFT (Dicrete-time Fourier Tranform d. (5 point Complete the miing entrie in the following table. Signal Fourier Tranform Characteritic Continuou in t & Fourier Serie Dicrete in ω Periodic Continuou in t Continuou-time Continuou in ω Fourier Tranform Dicrete in t Dicrete-time Continuou in ω & Periodic Fourier Tranform Dicrete in t & Periodic Dicrete Fourier Tranform Dicrete in ω & Periodic e. (5 point Show the internal tructure of a -point FFT. Make ure you label the input, output and multiplication factor clearly.

6 3. (5 point A linear ytem i decribed by the following tranfer function: u & & ( y a (8 point Write down the A, B, C, D matrice in the tandard form correponding to the above equation., (,, D C B A b (7 point Write down the tranfer function H(. ( ( ( ( 6 5 ( ( ( ( D B A I C H D C B A c ( point If the ame et of tate-variable equation alo repreent the following circuit, which phyical quantitie of the circuit do the tate variable, repreent? You mut jutify your anwer. Baed on the output equation, we can eaily deduce that i the voltage acro the F capacitor. By etting a the inductor current, we can write KVL around the bigger loop: w & KVL around the maller loop: & which are the ame a the tate equation. ½Ω ½H - ¼Ω F y(t - w(t

7 . (5 point An alternative to the impule-invariant technique i tep-invariance ynthei. The tep-invariant filter i derived by placing a unit tep on the input of an analog filter and a ampled unit tep on the input to a digital filter. The digital filter H(z i computed o that the output of the digital filter repreent ample of the output of the analog filter. y a (t Sampling T. y a (nt t Sampling T. H(z y d (nt a (5 point Write down the Laplace tranform of y a (t. Since the Laplace tranform of the tep function i /, we have L[ y a ( t] ( b ( point Compute the time-domain output y a (nt of the analog filter and then it Z- tranform Y a (z. L[ y a ( t] ( A B y y ( nt a Ya ( z z a ( t nt e z e t n nt e t t c (5 point Write down the Z tranform of the output Y d (z in term of H(z H ( z z Y d ( z d (5 point By equating Y d (z and Y a (z, compute H(z H ( z e z ( z

8 EXTRA WORKSHEET

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}