ECE 351 - Linear Systems II Homework #7 1. Find F(z), the z-transform of each of the following sequences: a. fk= 5 + 212!"#$%&k− 3 −4( )k()**+,--uk Answer: F(z) =5zz −1+2zz −12−3zz + 4 b. fk=12!"#$%&k − 2uk − 2 Answer: F(z) =1z z −12"#$%&' c. fk=12!"#$%&kuk − 2 Answer: F(z) =14z z −12"#$%&' d. fk= k314!"#$%&kuk Answer: 14z3+14z2+164zz −14"#$%&'4= F(z) 2. Find yk, the inverse z-transform for each of the following: a. Y (z) =6z3+ 29z2− 17z6 z +12"#$%&'z −13"#$%&'z − 1( ) Answer: yk= −4 −12"#$%&'k+ 213"#$%&'k+ 3()**+,--uk b. Y (z) =9z2− z − 2z2− 1 Answer: yk= 2δk+ 4 −1( )k+ 3"#$%uk c. Y (z) =−3zz2− z − 2 Answer: yk= − 2( )k+ −1( )k"#$%uk d. Y (z) =6z4+ 18z3+ 16z2− 8zz − 1( )z + 1( )3 Answer: yk= 3k2− 7k + 2( )−1( )k+ 4"#$%uk e. Y (z) =2z2+ 5z + 8z + 2( )2 Answer: yk= 2δk+32k −2( )kuk f. Y (z) =4z3+ 6z2z + 2( )z2+ 2z + 2( ) Answer: yk= 2 −2( )k+ 2 2( )kcos3πk4"#$%&'− 4 2( )ksin3πk4"#$%&'()*+,-ukg. Y (z) =3z3− 6z2− 8z z2+ 4( ) Answer: yk= −2δk−1+ 3 2( )kcosπk2"#$%&'uk− 2 2( )ksinπk2"#$%&'uk 3. Solve the following difference equations using Z-transforms. a. yk + 2+ yk +1+ yk= 0; y0= 1, y1= −2 Answer: yk= cos2πk3!"#$%&− 3 sin2πk3!"#$%&()*+,-uk b. yk + 2+ 3yk +1+ 2yk= 2 −3( )kuk; y0= 1, y1= 0 Answer: yk= 3 −1( )k− 3 −2( )k+ −3( )k"#$%uk c. yk+12yk −1= uk −1−12uk − 2; y−1= 0 Answer: yk=δk+13−43−12"#$%&'k()**+,--uk Read and perform the exercises in MATLAB Tutorial #7 before attempting problems 4 - 6 . For each problem turn in your MATLAB commands along with the results. 4. Use the residuez command in MATLAB to confirm the partial fraction expansion in problem 2a. 5. Use the residuez command in MATLAB to confirm the partial fraction expansion in problem 2b. 6. Use the residuez command in MATLAB to confirm the partial fraction expansion in problem