Signals

1. Some signal questions:
   1. Under what conditions is $a^n u[n]$ bounded?
   2. Any signal $x(t)$ can be decomposed into the sum of an even-symmetric signal $x_e(t)$ and an odd-symmetric signal $x_o(t)$, where $x_e(t) = 0.5(x(t) + x[-n])$ and $x_o(t) = 0.5(x(t) - x[-n])$. These are usually repesented as ${\mathcal{Ev}}(x(t))$ and ${\mathcal{Od}}(x(t))$ respectively. Given a signal $x(t) = \cos(2\pi t)u(t)$, determine if ${\mathcal{Ev}}(x(t))$ is periodic. If so, what is its period? Is ${\mathcal{Od}}x(t)$ periodic, and if so, what is its period?

   Systems

2. Systems are characterized according to the following properties:
   1. Memoryless
   2. Time-invariant
   3. Linear
   4. Causal
   5. Stable

   Which of the above properties do the following systems have:

   1. $y(t) = e^{x(t)}$
   2. $y[n] = x[n]x[n-1]$
   3. $y(t) = x(t/2)$
   4. $y[n] = x[2n]$
   5. $y[n] = \begin{cases} x[n]~~n\geq 1\\ 0~~n = 0 \\ x[n+1]~~n \leq -1 \end{cases}$

   Discrete time Fourier analysis

   Please answer any 5 of the following 6 questions:
3. Fourier series of periodic signals:
   1. Compute the Fourier series coefficients of $\cos(0.2\pi n) + \sin(0.3\pi n)$ (Hint: What is the period of sum? It is not the period of either the cosine term or the sine term).
   2. Does the signal $\cos(0.4 n)$ have a Fourier series expansion? Explain your answer?
4. What is the Fourier transform of $\cos(0.4 n)$? (Hint: Euler's formula).
5. The discrete-time Fourier transform of a signal $x[n]$ is $X(\Omega)$. We would like to define a new sinal $y[n] = x[\frac{an}{b}]$.
   1. What are the conditions on $a$ and $b$ to be able to define $y[n]$? (Hint, $x[\frac{an}{b}]$ is only defined when $\frac{an}{b}$ takes integer values.
   2. $y[n]$ cannot be defined for all values of $n$ as given above, since $\frac{an}{b}$ may not be integers for some values of $n$. Assuming $a$ and $b$ are integers, can you specify a complete form for $y[n]$? (Hint: For $a = 1$, we can define $y[n] = \begin{cases}x[n/b]~\mathrm{if}~n = kb\\ 0~otherwise\end{cases}$. Similarly, for $b=1$ we can define $y[n] = x[an]$ for any $n$).
   3. Specify the discrete-time Fourier transform of $y[n]$, $Y(\Omega)$, in terms of $X(\Omega)$, the discrete-time Fourier transform of $x[n]$.
6. Properties of Fourier transforms
   1. If $X(\Omega)$ is the discrete time Fourier transform of $x[n]$, show that the DTFT of $x[-n]$ is $X(-\Omega)$.
   2. If $x[n]$ is imaginary, show that the DTFT of $x^*[n]$ is $-X(\Omega)$.
   3. If $x[n]$ is real and symmetric, show that $X(\Omega)$ is also real and symmetric.
7. Some examples of DTFT
   1. Compute the DTFT of $a^{|n|} sin(\Omega_o n),~ |a|<1$.
   2. Compute the DTFT of $2^{-n} u[n]$.
   3. Compute the DTFT of $2^{-n} u[3-n]$. Use the solution to the problem above and the properties of the DTFT.
   4. Compute the inverse DTFT (i.e. the signal) for $X(\Omega) = \cos^2 \Omega$ (Hint, use the modulation property -- the transform of a product is the convolution of the transforms).
8. $x[n] = \cos(0.2\pi n)$. Plot the discrete time Fourier transform of
   1. $y[n] = x[n]$
   2. $y[n] = x[n/3]$
   3. $y[n] = \begin{cases} x[2n/3]~~\mathrm{if} n= 3k,~-\infty < k < \infty \\ 0 ~~~ otherwisee\end{cases}$

   Inverse Fourier Transforms

9. All four of the following are mandatory. Note that all the frequency responses are symmetric functions of frequency, so the impulse responses you compute should be real; if not your answer is wrong.
   1. The frequency response of a continuous-time low-pass filter is given by \[ H(\omega) = \begin{cases} 1,&~~|\omega| \le \omega_{max} \\ 0&~~otherwise \end{cases} \]

      What is its impulse response $h(t)$?

   2. The frequency response of a continuous time bandpass filter is given by \[ H(\omega) = \begin{cases} 1,&~~\omega_{min} \le |\omega| \le ~\omega_{max} \\ 0,&~~otherwise \end{cases} \]

      What is its impulse response $h(t)$? Note that according to the above specification, the frequency response actually has two pass bands: $-\omega_{max} \le \omega -\omega_{min}$ and $\omega_{min} \le \omega \le \omega_{max}$. Nevertheless, you should be able to build upon the solution to IX-1 using the frequency shift property.

   3. The frequency response of a discrete time low-pass filter is given by \[ H(\Omega) = \begin{cases} 1,&~~|\Omega| \le \Omega_{max} \\ 0&~~\Omega_{max} < |\Omega| \le \pi \\ H(\Omega + 2\pi) \end{cases} \]

      What is the impulse response $h[n]$ of the system?

   4. The frequency response of a discrete time band-pass filter is given by \[ H(\Omega) = \begin{cases} 1,&~~\Omega_{min} < |\Omega| \le \Omega_{max} \\ 0&~~0 \le |\Omega| \le \Omega_{min} \\ 0&~~\Omega_{max} < |\Omega| \le \pi \\ H(\Omega + 2\pi) \end{cases} \]

      What is the impulse response $h[n]$ of the system? Note that you can use the frequency shift property.