Homework Clarifications and Hints

Hmwk 1

• Sittler's Problem 1.2
  You are supossed to say whether the system is Linear and/or Time Invariant. Also, it is important to ignore the homogeneous solutions to the system in order for the system output to be both, "well defined" and LTI.
  You do not have to derive the solutions of the differential equations, just use the differential equations to prove or disprove the linear and time invariant properties.
• Sittler's problem 2.2
  To verify the particular solution, just plug it into the difference equation and verify that it holds. You don't have to derive the solution from the difference equation.
• O&S Problem 2.26 To find whether a signal could be an eigenfunction of a stable LTI system, use the fact the output can be expressed as the convolution of the input and its impulse response. Work the convolution sum out to see if it can be reduced to the product of the input and a term that is independent of n.
• O&S Problem 2.27
  System B
  To find whether there is another LTI system that can give you the same input-output pair, first observe that the system has to be causal, i.e. its impulse response h(n) must be zero for negative time indexes. Then Based on the fact that it is causal and the input-output pair given, show that h(0)=2, with this new information prove that h(1)=0 and so on. In this way you can prove that there is only one LTI system that gives you that input-output pair.
  System C
  To find whether there is another LTI system that can give you the same input-output pair, observe that the input for that problem is an eigenfunction and that the output its twice the input. Hence, if the system is LTI, you know its frequency response for w=1/8 is 2. However the frequency reponse for other frequencies is not specified and can be anything.
• O&S Problem 2.36
  Part a) There are many ways to solve ths prblem, but the basic idea in all of them is to use a contradiction argument, for example: Express a linear combination of shifted-in-time versions of x1, x2 and/or x3 as a linear combination of x1, x2 and/or x3. Using linearity you can find the output of the linear combination. Then assuming that the system is time invariant you can find the output using the linear combination of shifted versions and compare it to the output found using linearity. If the outputs do not match then the time-invariance assumption must be false so the systen is not time invariant.

Hmwk 2

• Sittler's Problem 4.1
  Please specify the ROC of the z-transforms you find.
  • d) consider the series expansion of e^z
  • e) and f) Use the differentiation technique (see Problem Session 2 Handout)
• Sittler's problem 4.3
  
  • a) Divide numerator and denominator by z^2 and use the upsampling property (ps file) (pdf file)
  • b) Factorize denominator and one of the poles will cancel with the zero.
  • c)Express the denominator as the square of a (1-az^(-1)) term and apply the differentiation technique (see Problem Session 2 Handout)
  • d) Use a series expansion or use the differentiation technique (see Problem Session 2 Handout)
  • e) Use the series expansion of sin(z)
  • f) Divide numerator and denominator by z^2 and use the upsampling property (ps file) (pdf file).
    The condition f(n) -> 0 as n -> infinity and as n-> -infinity does not imply that the sequence has to be double sided, it just implies that the sequence has to converge to zero on both sides. Notice for example that a geometric sequence like (1/2)^n u[n] satisfies that condition.
• Sittler's problem 4.4
  Recall that the convolution of two causal sequences is causal as well.

Hmwk 3

• Sittler's problem 6.1
  For this problem, see Problem Session #3 Handout.
• Sittler's problem 6.2
  Find a flow graph representation with only delays, to do that easily, first use a change of variable on "n" for each of the equations so that you only have difference equations with "n-k" or "n" -like terms.

Hmwk 4

• Sittler's Problem 9.1
  alpha is a single random variable, if it is chosen to be one, it is one for every f[n]. In other words , you only have two possible ensemble signals: f[n] = (-1)^n or f[n] = (-1)^(n+1).
  To obtain phi_ff(k), notice you only have two ensemble signals, so for example, for the first case alpha being zero or one with probability 1/2, we would have phi_ff(1) = (1/2)*(1)*(-1) + (1/2)*(-1)*(1).
• Sittler's Problem 9.2
  First do part b) (uncorrelated assumption) and then part a) (independence assumption) remembering idependence implies uncorrelation.
• Sittler's Problem 9.3
  Note: from the fact phi_ff(k) = delta(k) and that f(n) consists of uncorrelated samples, we can infer that f(n) is zero mean.
• Sittler's Problem 9.4
  Typo: phi_ff[n] should be phi_ff[k]
• Sittler's Problem 10.1
  The solution of this problem needs a very similar work as the one done in pages 46-47 from prof. Widrow's class notes. Take a look at it and work out what you would get if you had g_1(n) and g_2(b) insted of g(n) only.
• Sittler's Problem 10.2
  See Problem Session 4 Handout.

Hmwk 5

• Sittler's Problem 11.1
  
  • a) You can just show that ho[n] satisfies equation B from page 109 in Sittler's notes. You don't have to obtain ho[n] from that equation, just take the ho[n] given in the problem and show it satisfies the equation.
  • b) When it asks about how could a signal with the same autocorrelation as f[n] be simulated using a table of random numbers, by random numbers it refers to a D.T. random process with autocorrelation equal to delta[k], i.e., a process with zero mean and uncorrelated samples.
    Since the table of randoms numbers represents a signal with autocorrelatoin function delta[k], notice that if you pass it through a filter with impulse response h[n], the output will have an autocorrelation h[k]*h[-k]. So the idea is to find a impulse response h[k] such that h[k]*h[-k] = phi_ff[k].
• Sittler's Problem 11.2
  The message signal (and desired signal as well) the problem refers to, is the same as f[n] in problem 11.1 a)
  Besides the transfer function, you are supposed to find the impulse response as well.
  
• Sittler's Problem 12.3
  phi_ss(z)=5/3-z/2-z^-1/2 is factorizable, try to factor it in the form: phi_ss(z)=A(1-az^-1)(1-az); by equating 5/3-z/2-z^-1/2 and A(1-az^-1)(1-az) you will obtain two equations on A and a, which you can solve, they will give you two possible values for the pair a,A but only one pair of them will allow you to find phi_ss+(z), since phi_ss+(z) has to have its poles AND zeros inside the unit cricle. Note for example that an expression like 1-3z has its zero at 1/3 but a pole at infinity, so it cannot be phi_ss+(z).
  Besides the transfer function, you are supposed to find the impulse response as well.

Hmwk 6

• Problem 1
  Hints for problem 1 (ps file) (pdf file).
• Problem 2
  The overall transfer function of the system has only one pole which is a linear function of K.
• Problem 3
  For part b), it may be easier for you to find the output directly in the time domain.

Hmwk 7

• Problem 2 (O&S 4.3) part b)
  You don't have to give the solution given in the solutions on the Textbook, any valid T is ok, if you are curious to get that one, recall that cos(t) is an even function.
  
• Problem 6 (O&S 4.22) part c)
  The question asks for a range of values of T, but this is little bit misleading, since the answer consists of a range of values of T together with a specific value of T.
• Problem 7 (O&S 4.25) part c)
  In problem 4.25 from O&S (2nd Edition) in Figure P4.25-2, the left graph shows the transfer function of the reconstruction filter H_r(jOmega). For part c), make the height of the rectangular function equal to T instead of the fixed value shown there.

Hmwk 8

• Problems 1 and 2. Notice that the spectra of the output of the DAC is a train of impulses multiplied by the Transfer Function of the DAC. In the time domain what you have is a sum of sinusoids at different frequencies. In this hmwk you have to compute the power of the sinusoids at the specified frequencies. You can do that directly from the transform domain, since the power of a sinuoid is the sum of squared magnitude of the coefficient of its corresponding dirac deltas in the Fourier domain (Using Hertz). If you are using the rad/s units for the Fourier transform, you need to first divide the coefficients by 2*pi before squaring their magnitude.
• The solutions to all O&S problems are in the back of the book, so you can verify your answers
• Problem 3 (O&S 8.1) part c)
  The solution in the book is wrong, the 2\pi should be a 6.
  
• Problem 5 (O&S 8.7)
  To obtain x_1[n], take the 4-point IDFT of X_1[k].