Questions And Answers

In this page, you can ask questions and I will try to answer them. You can participate in
the discussion since this is a wiki. I will post some questions that students ask me after
class or during my office hours. Notice that whenever you write a question here, nobody
will know who wrote the question, so feel free to write questions.

Question: What are examples of some wide sense stationary processes ?

Reply: I gave one example in class. Here are a few others.

  • $X(t) = A \cos(2 \pi f_c t + \theta)$, where $A,f_c$ are constants and $\theta$ is uniformly distributed between $[0,2\pi)$
  • Let $X(t)$ be an i.i.d process and $Y(t) = X(t) + X(t-2)$. Notice that $Y(t)$ is not i.i.d
  • Let $X(t)$ be an i.i.d process and $Y(t)$ is any linearly filtered version of $X(t)$

Question: The baseband signal $s_l(t)$ is a complex signal in general. Didnt you say in class that voltage waveforms are real. So what is the meaning of a complex waveform ? I am confused.

Reply: There is no complex waveform physically. If you look at the quadrature implementation of the passband signal, i.e. think of the passband signal as

\begin{align} s(t) = x(t) \ \cos(2 \pi f_c t) - y(t) \ \sin(2 \pi f_c t) \end{align}

what this means is that we can take two real baseband waveforms $x(t)$ and $y(t)$ and independently modulate $\cos(2 \pi f_c t)$ and $\sin(2 \pi f_c t)$ thereby forming the passband signal. For mathematical convenience, we can think of a complex baseband waveform $s_l(t) = x(t) + j y(t)$ such that $s(t) = Re[s_l(t) \ e^{j 2 \pi f_c t}]$. Thus the complex notation is purely for mathematical convenience, there is always a bunch of real waveforms that get manipulated. It is nicer and simpler to represent it in complex notation. For example, if there is a phase shift $\theta$ introduced to the passband waveform, this simply corresponds to multiplying the complex baseband signal by $e^{j \theta}$. If we did not use the complex notation, then we would have to keep track of what happens to $x(t)$ and $y(t)$ separately.

This is somewhat similar to performing complex arithmetic on a computer. For example, in MATLAB you can multiply two complex numbers $A = A_I + j A_Q$ and $B = B_I + j B_Q$. Does it mean that the computer actually stores complex numbers? The computer only stores two real numbers corresponding to the real and imaginary parts. When we multiply the two numbers, the computer actually multiplies the real numbers, i.e. If $C = A B = C_I + j C_Q$, then $C_I = A_I B_I - A_Q B_Q$ and $C_Q = A_I B_Q + B_I A_Q$. However, it would be a pain to keep track of $C_I$ and $C_Q$ separately everytime, the notation is so much easier if we simply thought of $C = A B$

Question: While representing a bandpass signal as a low pass signal, as we see in the frequency domain, we are effectively taking only half the power of the original signal. Why is it that mathematically we actually do not get exact half?


Question: In problem 6 of the homework, the given signal S(t) is the base band signal …. isn't it? Also, it is a sinc function in the time domain which is symmetric around the y-axis in the frequency and thus shouldn't have any quadrature component without any conditions. Is that true?

Answer: Yes, then your answer is that $f_0 = 0$. Can you find a non-zero $f_0$ such that the quadrature component is zero. Notice that if you choose an arbitray $f_0$, it is not true that the quadrature component is zero (I dont quite understand what you mean by without any conditions, but it is clearly not true for all $f_o$). For example, you can also think of this signal as $\frac{1}{1024 \pi t} \ \sin(2 \pi 512 t)$. Thus, if we choose $f_0 = 512$ you get only a quadrature component and the in-phase is zero!

Question: Regarding the above answer, isn't there a big assumption that the signal needs to be narrow band around the carrier frequency $f_0$ when we talk about the complex baseband equivalent represenation? If we choose $f_0=512$, but the signal bandwith is from -512 to 512 Hz, is it still appropriate to talk about the signal's in-phase and quadrature components?

Question: What is meant by the complex envelope? Is it a(t) exp (j Theta(t)) where a(t) is the real envelope and Theta(t) is the excess phase?

Answer: Yes you are correct. It is just the baseband signal. It is called the complex envelope because this baseband signal modulates a carrier with much higher frequency and hence the envelope of the resulting high frequency waveform will be determined by the baseband signal.

Question: Does anyone know if we have a class today Feb 6th or not? When and where?

Answer: I sent an email earlier today. There is no class. I will post video lectures for this class. You can watch them whenever you feel like.
Check the site later today or wait for my email. - Krishna

Question: There seems no class notes for signal space review.
Comment: I agree, I am having difficulties accessing the videos and notes files. The first video gives a 403 - Forbidden error and the notes do not open. Further inspection shows an inaccuracy in the link for those files (http://// and not the usual http://). Manual correction of the link does not solve the problem though. Is anyone else experiencing this issue?

Answer: I have fixed this link now. Please try again.

Question: This is really more of a commnet. Proakis defines the cross correlation as $\phi_{xy}(\tau) = E[x(t+\tau) y(t)]$, not the standard way that I know of as $\phi_{xy}(\tau) = E[x(t) y(t+\tau)]$. This is not explicitly stated in the text (I don't think) but his mathematical formulation (if you look carefully) uses this definition. The class notes reproduce whats in the text so they too follow this definition. However there is one problem in the class notes and that is in the deffinition of the autocorrelation of $z(t)$. The text defines $\phi_{zz}(\tau) = \frac{1}{2}E[z^*(t) z(t+\tau)]$, but the notes have $\phi_{zz}(\tau) = \frac{1}{2}E[z(t) z^*(t+\tau)]$. The notes however produce the results from the text which are accurate acording to the books definition, but if you followed the way $\phi_{zz}(\tau)$ is defined in the notes you will get a slightly different result. I redid the entire derivation of everything from page 5 to 7 of lecture 4 notes using what I know to be the standard way of defining the cross correlation. The only difference that arises is that in my derrivation I have $\Phi_{xy}(f)$ where ever the text/notes have $\Phi_{yx}(f)$. I don't think this is a huge deal, but I think it is a minor discrepency that should be brought forth incase anyone is having trouble with the derivations.

The only thing I am a little unsure of is at the very end of page 7. The notes have

\begin{eqnarray} \lefteqn{ h(f) = j\Phi_{yx}(f) = L.P. \{\Phi_{NN}(f+f_c) - \Phi_{NN}(f-f_c)\} } \\ & & or\hspace{3 mm} \Phi_{xy}(f) = j L.P. \{\Phi_{NN}(f+f_c) - \Phi_{NN}(f-f_c)\} \end{eqnarray}

In my derrivation under the standard definition of the cross correlation I get that

\begin{eqnarray} \lefteqn{ h(f) = j\Phi_{xy}(f) = L.P. \{\Phi_{NN}(f+f_c) - \Phi_{NN}(f-f_c)\} } \\ & & or\hspace{3 mm} \Phi_{yx}(f) = j L.P. \{\Phi_{NN}(f+f_c) - \Phi_{NN}(f-f_c)\} \end{eqnarray}

Is everything ok here?

Question: Do we have class notes on last Wednesday's lecture to download?

Question: In question 2 of hw 2, I have a doubt in the solution. We selected by inspection four orthonormal basis vectors and stated that the dimensionality is 4. But I think that from the inspection we can only state that dimensionality < = 4 and exact equality can only be derived from Gram Schmidt. Is this logic right??

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