Review Problems 1 for EE5350, 2014 Fall 9/22/2014

Closed Form Problems

1. h(n) = cos(won)u(n), x(n) = u(n), y(n)=h(n)*x(n). Find the closed form of y(n).

2. Find the approximation of the sum: ( )∑=

4

0

2.0n

n

3. Find the closed form for ( )∑∞

=

−0

2

n

nyx

4. Find a closed form expression for ( )∑=

M

n

nx0

sin2

5. Find H(ejw) in closed form for the system described by the recursive difference equation

( ) ( ) ( ) ( ) ( )15

752

15

11

15

8 −−+−−−= nxnxnynyny

6. Find the closed form for ∑−=

−20

5

3

k

kxe

7. Find the closed form for ( )∑∞

=0

32sink

kxnx

8. Find the closed form for ( )( )∑=

−29

50

m

mxJ

9. The impulse response of a moving-average system is [ ]

≤≤−++=

otherwise

MnMMMnh

,0

,1

121

21

Express the frequency response in closed form

10. Find the DTFT of [ ] [ ]nuanx n= , in closed form.

Convolution Problems

1. h(n) is non-zero for N0 ≤ n ≤ N1, x(n) is non-zero for N2 ≤ n ≤ N3, y(n) non-zero for N4 ≤ n ≤ N5. Give N0,N1,N2,N3 , find N4 and N5

( ) ( ) ( )∑=

−=1

0

N

Nk

knxkhny

2. Convolve h(n) and x(n) to get y(n). Put y(n) in closed form when possible.

h(n) = 3-nu(-n) and x(n) = 2-nu(-n).

3. Convolve h(n) and x(n) to get y(n). Put y(n) in closed form when possible.

h(n) = δ(sin((2π/N)n) and x(n) = n2 where N is odd.

4. Convolve h(n) and x(n) to get y(n).

h(n) = δ(sin(1 + |n|)) and x(n) = sin(n2) .

5. Convolve h(n) and x(n) to get y(n).

h(n) = u(n+1)-u(n-5), x(n) = u(n+1)-u(n-4). Express the result in terms of r(n), where u(n)*u(n) = r(n+1).

6. Convolve h(n) and x(n) to get y(n).

( ) ( ) ( )12 −−= nnnx δδ

( ) ( ) ( ) ( )212 −+−+−= nnnnh δδδ

7. Convolve h(n) and x(n) to get y(n).

h(n) = δ(n-no) - δ(n-n1) and x(n) = exp(-n2)u(n).

8. Convolve h(n) and x(n) to get y(n).

h(n) = δ(n-n1) - δ(n-n3) and x(n) = sin(n2).

9. Convolve h(n) and x(n) to get y(n).

h(n) = anu(n) and x(n) = anu(n) for real “a”. The y(n) expression should include r(n).

10. Convolve h(n) and x(n) to get y(n).

h(n) = δ(n-n1) - δ(n-n3) and x(n) = cosh (n2).

11. Develop a general expression for the output y[n] of an LTI discrete time system in terms of its input x[n] and the unit step response s[n].

12. A periodic sequence x[n] with a period N is applied as an input to an LTI discrete time system characterized by an impulse response h[n] generating an output y[n]. Is y[n] periodic? If yes, what is the period?

13. Let, 1 2[ ] [ ]* [ ]y n x n x n= and 1 1 2 2[ ] [ ]* [ ]z n x n N x n N= − − . Express z[n] in terms of y[n].

14. Let y[n] = x[n]*h[n]. Prove that

[ ] ( [ ]) ( [ ])n n n

y n x n h n= ×∑ ∑ ∑

15. Convolve h(n) and x(n) to get y(n). Put y(n) in closed form when possible.

(a) h(n) = an u(n) and x(n) = bn u(n).

(b) h(n) = cnu(n), x(n) = cnu(n).

(c) h(n) = u(n+5)-u(n-7), x(n) = u(n+1)-u(n-8). Express the result in terms of r(n) where

u(n)*u(n) = r(n+1).

(d) h(n) = cos(wcn)u(n), x(n) = u(n).

16. Find the convolution of the given sequences in closed form

(a) u[n] and an u[n]

(b) u[n] and nan u[n]

17. The sequence of Fibonacci numbers f[n] is a causal sequence defined by,

f[n]= f[n-1]+ f[n-2], n≥2

With f[0]=0 and f[1]=1. Show that f[n] is the impulse response of a causal LTI system described by the difference equation :

y[n]= y[n-1]+y[n-2]+x[n-1]

18. Let, 1 2[ ] [ ]* [ ]y n x n x n= and 1 1 2 2[ ] [ ]* [ ]z n x n N x n N= − − . Express z[n] in terms of y[n].

Forward DTFT Problems

1. Determine the DTFT for the signal

[ ] ( )44

1 +

= nunxn

2. Determine the DTFT of the sequence

[ ]

=,0

,1nr

otherwise

Mn ≤≤0

3. Find the DTFT of the sequence

[ ] [ ]5−= nuanx n

4. 4.Give the DTFT pair anu(n)↔ ωjae−−1

1 1<a , use the Eq. to determine the DTFT X(ejw)

of the sequence.

[ ] [ ]−

=−−−=0

1n

n bnubnx

0

1

≥−≤

n

n

What restriction on b is necessary for the DTFT of x[n] to exist?

5. Consider the sequence [ ]1 2

1 cos , 02

0,

nn M

w n M

otherwise

π − ≤ ≤ =

Express W(ejω), the DTFT of [ ]nw , in terms of R(ejω), the DTFT of [ ]nr .

6. Find the forward transform of ( ) ( ) ( )1−+= nnnx δδ

7. Let the real discrete-time signal x[n] with DTFT( )ωjeX be the input to a system with the

output defined by

[ ] ( )

0

if n is evex ny n

other e

n

wis

=

Express ( )ωjeY the DTFT of the output, as a function of ( )ωjeX and ( )ωjeS

8. Give the DTFT pair x(n)↔ ωjae−−1

1. use the Eq. to determine the DTFT of the signal

x(2n+1)

9. Compute the DTFT of x[n]=u[n-2]-u[n-4].

10. Find the DTFT of the signal

[ ] 11, ≤≤−= aanx n .

11. Let h[n] be a real valued sequence such that |H(ejω)|=1 for all values of ω. Prove that the sequence h[n] is shift orthogonal i.e.

[ ] [ ] [ ]n

h n h n m mδ− =∑ .

12. A sequence x[n] has DTFT given by |X(ejω)|. Let y[n] is the downsampled (by 2) version of x[n] defined by, y[n] = x[2n]. Express |Y(ejω)| in terms of |X(ejω)|.

13. For a real sequence x[n] prove that , X(e-jω) = X*( ejω).

14. A digital filter h(n) is to be designed as h(n) = h1(n)⋅h2(n), where all three impulse responses are real.

(a) Express H(ejw) in terms of H1(ejw) and H2(e

jw)

(b) If H1(ejw) and H2(e

jw) have the same ideal lowpass amplitude response with cut-off frequency wc < π/2, and phases φ1(w) = φ2(w) = -.5(N-1)⋅w, sketch the magnitude response and give the phase response φ(w) of the filter H(ejw)

Difference Equation Problems With Frequency and Impulse Response

1. A linear constant-coefficient difference equation

[ ] [ ] [ ] [ ]14

11

2

1 −−=−+ nxnyxnyny

What are the impulse, frequency response for the causal LTI system satisfying this difference equation?

2. Determine the frequency response of the system whose input and output satisfy the difference equation

[ ] [ ] [ ] [ ] [ ]212121 −+−−=−+ nxnxnxnyny

3. If

[ ] [ ] [ ] [ ]131

261

165 −=−+−+ nxnynyny

(a)What are the impulse, frequency response for the causal LTI system satisfying this difference equation?

(b) What is the general form of the homogeneous solution of the difference equation?

(c) Consider a different system satisfying the difference equation that is neither causal nor LTI, but that has y[0]=y[1]=1. Find the response of the system to x[n]=δ[n].

4. Consider the difference equation representing a casual LTI system

[ ] [ ] [ ].111 −=−

+ nxnya

ny

(a) Find the impulse response of the system, h[n], as a function of the constant a.

(b) For what range of values of a will the system be stable?

5. A linear time invariant system is described by the recursive difference equation

(a) Find H(ejw) in closed form.

(b) Find the homogeneous solution.

(c) Find h(0) and h(1)

(d) Find the impulse response h(n).

6. Consider an LTI system defined by the difference equation.

[ ] [ ] [ ] [ ]22142 −−−+−= nxnxnxny

(a) Determine the impulse response of the system.

(b) Determine the frequency response of this system. Express your answer in the form

( ) ( ) dnjjj eeAeH ωωω −= ,

where ( )ωjeA is a real function of ω.

7. An LTI system has the frequency response

( ) ω

ω

ω

ωω

j

j

j

jj

e

e

e

eeH −

−

−

−

−−=

−−=

8.01

45.01

8.01

25.11

(a) Specify the equation that is satisfied by the input [ ]nx and the output [ ]ny .

(b) Use one of the above forms of the frequency response to determine the impulse response [ ]nh .

8. An LTI discrete-time system has the frequency response

( ) ( )( )ω

ω

ωω

ω

ω

ωωω

j

j

jj

j

j

jjj

e

e

ee

e

e

jejeeH −

−

−−

−

−

−−

−+

−=

−+=

−+−=

8.018.01

1

8.01

1

8.01

11 22

(a) Use one of the above forms of the frequency response to obtain an equation for the impulse response [ ]nh of the system.

(b) From the frequency response, determine the difference equation that is satisfied by the input [ ]nx and the output [ ]ny of the system.

13 5

56 6 6

1y(n) = x(n)- x(n -1)+ y(n -1)- y(n - 2)

9. A system is described by the recursive difference equation

Re-write the difference equation so that it generates the impulse response h(n). Give numerical values for h(0) and h(1). Find the impulse response h(n)

10. An LTI system is described by the input-output relation

[ ] [ ] [ ] [ ]212 −−−+= nxnxnxny

(a) Determine the impulse [ ]nh and frequency response ( )ωjeH1 of the system.

(b) Now consider a new system whose frequency response is ( ) ( )( )πωω += jj eHeH1 .Determine

[ ]nh1 ,the impulse response of the new system.

11. Determine a closed form expression for the frequency response H(ejω) of the causal LTI discrete time system characterized by,

y[n] = x[n]+ a y[n-R], |a|<1

Determine the minimum and maximum values of the magnitude response. How many peaks and dips occur in the range 0≤ω≤2π and what are their locations.

12. The trapezoidal integration formula can be represented by a difference equation given by,

y[n]= y[n-1]+0.5(x[n]+x[n-1]),

with y[-1]=0. Considering the system to be causal, determine the frequency response of the above system.

13. A causal LTI discrete time system is described by the difference equation,

y[n]- 0.6y[n-1] = x[n]

Determine the impulse response of the system.

14. The sequence of Fibonacci numbers f[n] is a causal sequence defined by,

f[n]= f[n-1]+ f[n-2], n≥2

With f[0]=0 and f[1]=1. Show that f[n] is the impulse response of a causal LTI system described by the difference equation :

y[n]= y[n-1]+y[n-2]+x[n-1]

5 1 5

y(n) = y(n -1) - y(n - 2) - x(n) + x(n -1)12 24 12

Parseval’s Equation

1. Use Parseval’s Equation to prove:

∑∞

==

12

2 12

3 n n

π

2. Assume that

(a) Find Y(ejw) in terms of X(ejw) and H(ejw) if

(b) Express the quantity E using H(ejw) and X(ejw) if

3. Let x[n] and y[n] denote complex sequences and )( ωjeX and ( )ωjeY their respective DTFT.

(a) By using convolution theorem and appropriate properties,determine in terms of x[n] and y[n], the sequence whose DTFT is ( ) ( )ωω jj eYeX ∗ .

(b) Using the result in part (a),show that

[ ] [ ] ( ) ( )∫∑−

∗∞

−∞=

∗ =π

π

ωω ωπ

deYeXnynx jj

n 21

Equation is a more general form of Parseval’s equation.

(c) Using the equation given in (b) determine the numerical value of the sum

| |wjwX( ) = e for | w| e π≤

n)-h(k)x(k = y(n)- = k∑

∞

∞

(k)x(k)h = E *

- = k∑

∞

∞

( ) ( )∑

∞

−∞=n n

n

n

n

ππ

ππ

56/sin

24/sin

4. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find frequency domain expressions for the following:

(c) Find the numerical value of

sin 8sin 5

n = -

( n/ )( n/ )

3 n 7 n

πππ π

∞

∞

− •∑

5. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find frequency domain expressions for the following;

(c) Find the numerical value of

ωπ

πωω d

ee jj∫−

−− −− 8.011

5.011

21

6. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find frequency domain expressions for the following;

(c) Find the numerical value of

sin 11sin 76 3n = -

( n / )( n / )

n n

πππ π

∞

∞

−− •∑

( )n = - k = -

a C = x(n) y(n) (b) y(n) = h(k)x(k n)∞ ∞

∞ ∞

⋅ −∑ ∑

( ) *n = - k = -

a C = x(n) y ( n) (b) y(n) = h(k)x( k n)∞ ∞

∞ ∞

⋅ − − −∑ ∑

( )n = - k = -

a C = x(n) y( n) (b) y(n) = h( k)x(k n)∞ ∞

∞ ∞

⋅ − − −∑ ∑

7. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find frequency domain expressions for the following;

(c) Find the numerical value of

sin 9sin 3

n = -

( n/ )( n/ )

3 n 7 n

πππ π

∞

∞

•∑

8. Let x(n) and y(n) denote complex sequences with DTFTs X(ejw) and Y(ejw).

(a) Find a frequency domain expression for the constant;

(b) Find the numerical value of

9. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find frequency domain expressions for the following;

(c) Using part (a), find the numerical value of

( ) *

n = - k = -

a C = x(n) (n) (b) y(n) = h(k)x(-k n)y∞ ∞

∞ ∞

• −∑ ∑

(n)yx(n) = C *

- = n

•∑∞

∞

n7

n/8)(

n3

n/2)(

- = n ππ

ππ sinsin •∑

∞

∞

(a) . Give the substitution you made for h(-n).

(b)

n = -

k = -

C = x(n) h( n)

y(n) = h(k)x(k n)

∞

∞

∞

∞

⋅ −

−

∑

∑

ωπ

π ωωd

ee jj∫

− −− −•

−61

1

1

21

1

1

2

1

10. A sequence has the DTFT

( ) ( )( ) .1,11

1 2

<−−

−= − aaeae

aeX

jjj

ωωω

Calculate ( ) ( ) ωωπ

π

π

ω deX j

∫−

cos2

1

11. Using Parseval’s relation evaluate the following integrals,

(a) 0

4

5 4cos

π

ω+∫

(b) 20

4

(5 4cos )

π

ω−∫

(c)

2| ( ) |jdX e d

d

πω

π

ωω−

∫

12. Let x[n] be a length 9 sequence given by,

x[n]=3 0 1 -2 -3 4 1 0 -1

with a DTFT X(ejω). Evaluate the following expressions without computing the DTFT X(ejω) itself.

(a) X(ej0)

(b) ( )jX e dπ

ω

π

ω−∫

(c) 2| ( ) |jX e dπ

ω

π

ω−∫

(d) - 2| ( ) |jdX e d

d

πω

π

ωω−

∫

Sampling and Reconstruction Problems, Including Aliasing or Decimation

1. A C/D converter samples xc(t) with a sampling period T to produce x(n). An ideal lowpass digital filter h(n), with cut-off frequency wo is applied to x(n) to produce y(n). A D/C converter generates yc(t) from y(n), again using the sampling period T. The cut-off frequency of xc(t) is 3 radians/sec.

(a) Find the cut-off frequency of x(n) in radians, as a function of T.

(b) Find the largest sampling period T such that y(n) has no aliasing. (Hint: x(n) may still be

aliased)

(c) Find the smallest sampling period T so that the filter h(n) modifies the spectrum of x(n).

(d) Give an expression for h(n).

2. The analog signal xa(t) is ideally sampled at a rate of 2π/T radians/sec., producing the discrete

time signal x(n), where T is the sampling period in seconds. We want to decimate x(n) with an

integer decimation rate N1 , so that the resulting signal y(n) has a sampling period of N1⋅T.

Assume that this decimation requires us to lowpass filter x(n) before sub-sampling. In sub-sampling, we will delete consecutive groups of (N1 -1) samples and keep every N1th sample.

(a) If the sampling period of y(n) is to be N1⋅T, what is its sampling rate in radians/sec ?

(b) If y(n) is not aliased, what is the maximum cut-off frequency of x(n) in radians/second?

(c) Multiplying your answer in part (b) by T, find the required cut-off frequency wo in radians for the lowpass decimation filter h(n).

(d) Give the impulse response h(n) in terms of n and N1

3. The DTFT, Xn(ejw), of an N-sample window ending at time n, can be written as

(a) If the most recent sample available is x(n), is the calculation of Xn(ejw) causal ?

(b) Given a value for w, and given that e-jw(m-n) is already pre-calculated, how many real multiplies are required to calculate Xn(e

jw) for x(m) real ?

(c) Give an efficient method for calculating Xn(ejw) from Xn-1(e

jw)

(d) How many real multiplies are required in part (c) if x(n) is real ?

4. The discrete time signal x(n) has a cut-off frequency that may be as large as π radians.

(a) Assuming that x(n) comes from ideal sampling of xa(t) at a rate of 2π/T radians/sec., with no

aliasing, give the highest possible Nyquist frequency ΩN for Xa(jΩ).

(b) If we decimate x(n) with an integer decimation rate N1 , so that the resulting signal y(n) has a

sampling period of N1⋅T, give the new sampling rate and the new Nyquist frequency in radians per

second.

(c) Given this new Nyquist frequency, what cut-off frequency must our lowpass anti-aliasing filter h(n) have in radians ? ( Remember, we have to lowpass filter x(n) before subsampling it to get y(n) )

(d) Give the impulse response for h(n), assuming it has a time delay of zero.

5. A signal x(n) has N samples numbered 0 to N-1. The pseudocode below should calculate y(n) by using the causal difference equation, y(n) = x(n) - .5 y(n-1). Assume that legal arguments n in x(n) and y(n) are numbered 0 to N-1.

y(A) = B

For n = C to D

y(n) = x(n) - .5 y(n-1)

End

(a) For the first statement, give B, and a legal value for A, so that y(C) in the third statement can be calculated.

(b) In the second statement, give values for C and D so that the samples y(n) are calculated in the proper order, using only legal values for n.

(c) Is the filter stable (yes or no) ?

6. A signal x(n) has N samples numbered 0 to N-1. The pseudoCode below should calculate the magnitude and phase responses from x(n) for M frequencies w(k), where k varies from 0 to M-1. These frequencies are evenly spaced and include 0 and π. The only complex variables are z and H.

∆w = A

w = -∆w

For 0 ≤ k ≤ M-1

w = w+∆w

H = B

z = e-jw

For 0 ≤ n ≤ N-1

H = H+C

End

Amp(k) = H

E = RealH

F = ImH

Phi(k) = G

ww(k) = w

End

(a) Give the value of A, so that w varies from 0 to π as k varies from 0 to M-1.

(b) Give values for B and C, so that samples of the frequency response are temporarily stored in the variable H.

(c) Give expression G in terms of E and F, so that the phase is stored as Phi(k).

7. For the infinite length sequence x(n), we want to calculate a frequency domain model in a moving N-sample window as

(a) Express Yn-1(k) as Yn(k)WNk plus two additional terms.

(b) Replacing n by n+1 in part (a), give a final expression for Yn(k) in terms of Yn+1(k)

8. A signal x(n) has an infinite number of samples and is to be convolved with a causal filter h(n), which is nonzero for 0 ≤ n ≤ N1-1. The output is y(n).

(a) The convolution can be done efficiently as follows. Let xm(n) = x(n+(m-1)⋅N2) for 0 ≤ n ≤ N2-1 and xm(n) = 0 for N2 ≤ n ≤ N3-1. Here, m varies from 1 to infinity. The impulse response array h(n) is padded with zeroes so that h(n) = 0 for N1 ≤ n ≤ N3-1. Now, Ym(k) = H(k)⋅Xm(k) and ym(n) = DFT-1Y m(k). Give the smallest possible DFT order, N3.

(b) Give the delay N5 so that y(n) from part (a) can be constructed as

(c) The filter h(n) is symmetric (even or odd) about sample number N6 (so h(N6-k)= h(N6+k) for example). Since y(n) is produced one block at a time, some samples are delayed more than others. Give the minimum and maximum time delays for samples of y(n), if y(n) samples are output a block at a time.

(d) Give the minimum allowable value for N2.

9. The spectrum (CTFT) Ma (f), with zero‐phase and Ma (0) = 1, of a continuous‐time

analog signal ma (t) is shown in the figure below. A discrete‐time signal x[n] is generated

by time‐sampling ma (t) at a sampling period of 10-4 second, where n denotes the sample

index.

(a) Sketch the spectrum (DTFT) of x[n], X(ejω) for -2π≤ω≤2π

(b) Determine the inverse DTFT of X(ejω) and propose a method to reconstruct ma (t )

using the time‐sampled signal x[n].

10. A causal, linear-phase bandpass FIR digital filter h(n) is to be designed with cut-off frequencies of .3 radian and 2.3 radians, and with a phase of -23w.

(a) Find an expression for hd(n) using the inverse DTFT.

(b) Find the filter's time delay in samples.

(c) If hd(n) is windowed to get h(n), find the largest possible value for N.

(d) Give the appropriate Hamming window for the length-N causal filter of part (c).

11. Consider sampling the signal

(a) Sketch the FT of the sampled signal for the following sampling intervals:

(i) Ts = 1/8

(ii) Ts = 1/3

(iii) Ts = 1/2

(iv) Ts = 2/3

12. A sequence x[n] has DTFT given by |X(ejω)|. Let y[n] is the down sampled(by 2) version of x[n] defined by, y[n] = x[2n]. Express |Y(ejω)| in terms of |X(ejω)|.

Inverse DTFT Problems

1. Evaluate the inverse DTFT for,

2. Evaluate the inverse DTFT for,

(a)

(b)

3. For |w| ≤ π, and a real value for d, assume that

(a) Find x(0).

(b) Find lim x(n) as n approaches infinity.

(c) Is x(n) even, odd or neither ?

(d) Find an expression for x(n).

4. A causal LTI discrete time system is described by the difference equation,

y[n]- 0.6y[n-1] = x[n]

Determine the impulse response of the system.

5. Evaluate the inverse DTFT for

H(ejω)= 1+cosω+cos2ω

6. A filter is needed to recover x1(n) from the signal, x(n) = x1(n) + x2(n) + x3(n), where x1(n) =

sin(.5n), x2(n) = sin(1.4n), and x3(n) = cos(2.7n).

(a) What kind of filter is required? LP, BP, HP or BR?

(b) Specify a filter cut-off frequency or frequencies in radians and find the corresponding cut-off sample or samples, k1 etc, for H(k).

(c) Assume that H(k) is ideal in the sense that each of its samples is either 1 or 0. Find a closed form for the impulse response h(n) using H(k), the inverse DTFT and the cut-off sample symbol(s) ki.

(d) Find a real expression for h(n) from part (c).

7. A bandpass FIR digital filter is to be designed with a cut-off frequencies of .5 radian and 2 radians, and with a phase of -25w.

(a) Find an expression for hd(n) using the inverse DTFT.

(b) Find the filter's time delay in samples.

(c) If hd(n) is windowed to get a causal, linear-phase filter h(n), find the largest possible value for N.

(d) Give the appropriate Hamming window for the length-N causal filter of part (c).

8. A causal, linear-phase bandpass FIR digital filter h(n) is to be designed with cut-off frequencies of .3 radian and 2.3 radians, and with a phase of -23w.

(a) Find an expression for hd(n) using the inverse DTFT.

(b) Find the filter's time delay in samples.

(c) If hd(n) is windowed to get h(n), find the largest possible value for N.

(d) Give the appropriate Hamming window for the length-N causal filter of part (c).

9. Determine the sequence y[n] whose DTFT is

10. Show that the inverse DTFT of

1( ) , |a| < 1

(1 )j

j mX e

aeω

ω−=−

is given by,

( 1)![ ] [ ]

!( 1)!nn m

x n a u nn m

+ −=− .

11. Evaluate the inverse DTFT for,

(a) ( ) ( 2 )j

k

X e kω δ ω π∞

=−∞= +∑

(b) 0

( ) 1 2 cos( )N

j

l

Y e lω ω=

= + ∑

12. Evaluate the inverse DTFT for,

H(ejω)= 1+cosω+cos2ω

13. Without computing the IDTFT determine which of the following DTFT’s have an inverse that is an even sequence, and which has an inverse that is an odd sequence:

a. X(ejω)= jω for 0≤|ω|≤π

b. X(ejω) = |ω| for 0≤|ω|≤ωc

= 0 for ωc ≤|ω|≤π

Recursive Filter or DTFT Calculation Problems

1. A system is described by the recursive difference equation

(a) Find H(ejw) in closed form. Give H(ej0) and H(ejπ).

(b) Give the homogeneous solution to the difference equation above.

(c) Re-write the difference equation so that it generates the impulse response h(n).

Give numerical values for h(0) and h(1).

(d) Using your answers to parts (b) and (c), give the impulse response h(n).

(e) Is the system stable ? (Yes or No)

2. Consider a causal LTI system that is cha3racterized by the difference equation

(a) What is the impulse response?

(b) If the input to this system is , what is the system response to this input

signal?

3. Let x(n), h(n) and y(n) denote complex sequences with DTFTs X(ejw), H(ejw) and Y(ejw). Find

frequency domain expressions for the following;

4. Find the DTFT representations for the following periodic signal: Sketch the magnitude and phase spectra.

5. Compute the discrete-time Fourier transform of the following signals.

(a)

(b) ,

6. Consider the linear constant coefficient difference equation

y[n] – 0.5y[n - 1] = x[n]

which describes a linear, time-invariant system initially at rest. What is the system function that describes Y(Ω) in terms of X(Ω)?

7. A discrete-time causal, linear, time-invariant system is described by the following input-output equation:

Here, x is the input to the system and y is the corresponding output.

(a) Find the frequency response of this system.

(b) Find the impulse response of this system.

(c) Find the response of this system to the input signal x1[n] = δ[n] − (1/3)δ[n − 1].

(d) Find the response of this system to the input signal x2[n] = ejπn

8. Consider the following discrete –time signal

(a) Does the signal have even or odd symmetry?

(b) Does this signal primarily have mid, low or high frequency components?

(c) Find the DTFT of the signal.

(d) Plot the signal. Also plot the real and imaginary parts over the range of -5π to 5π

(e) What is the fundamental period of the DTFT?

9. Suppose we have an LTI system characterized by an impulse response

(a) Sketch the magnitude of the system transfer function.

(b) Evaluate y[n] = x[n] * h[n] when

10. The DTFT, Xn(ejw), of an N-sample window ending at time n, can be written as

(a) If the most recent sample available is x(n), is the calculation of Xn(ejw) causal ?

(b) Given a value for w, and given that e-jw(m-n) is already pre-calculated, how many

real multiplies are required to calculate Xn(ejw)for x(m) real ?

(c) Give an efficient method for calculating Xn(ejw) from Xn-1(e

jw)

(d) How many real multiplies are required in part (c) if x(n) is real ?