1

(Difficulty: $\star$ ) Among the choices below, select all the linear systems. (Please note that some of the choices use functions rather than discrete-time signals; the concept of linearity is identical in both cases).

• Second derivative, i.e.
$y(t)=\frac{d^{2} }{d t^{2} } x(t)$

• The DTFT, i.e. transform a sequence $\mathbf{x}$ into $\text{DTFT} \{\mathbf{x}\}$

• Scrambling, i.e. a permutation to the input sequence, e.g.:

• AM radio modulation, i.e. multiply a signal $x[n]$ by a cosine at the carrier frequency:
$y[n]=x[n] \cos \left(2 \pi \omega_{c} n\right)$

• Clipping, i.e. enforce a maximum signal amplitude $M, e . g:$
$y[n]=\left\{\begin{array}{ll}x[n] & , x[n] \leq M \ M & , \text { otherwise }\end{array}\right.$

• Envelope detection (via squaring), i.e. $y[n]=|x[n]|^{2} * h[n]$ ,where $h[n]$ is the impulse response of a lowpass filter such as the moving average filter.

• Time-stretch, i.e. $y(t)=x(\alpha t)$ ,e.g. if you play an old $\mathrm{LP}-45$ vinyl disc at $33 \mathrm{rpm}$, the time-stretch coefficient would be $\alpha=33 / 45$

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2

(Difficulty: $\star$ ) Among the choices below, select all the time-invariant systems. (Please note that some of the choices use functions rather than discrete-time signals; the concept of time invariance is identical in both cases).

• Second derivative, i.e.
$y(t)=\frac{d^{2} }{d t^{2} } x(t)$

• The DTFT, i.e. transform a sequence $\mathbf{x}$ into $\text{DTFT} \{\mathbf{x}\}$

• Scrambling, i.e. a permutation to the input sequence, e.g.:

• AM radio modulation, i.e. multiply a signal $x[n]$ by a cosine at the carrier frequency:
$y[n]=x[n] \cos \left(2 \pi \omega_{c} n\right)$

• Clipping, i.e. enforce a maximum signal amplitude $M, e . g:$
$y[n]=\left\{\begin{array}{ll}x[n] & , x[n] \leq M \ M & , \text { otherwise }\end{array}\right.$

• Envelope detection (via squaring), i.e. $y[n]=|x[n]|^{2} * h[n]$, where $h[n]$ is the impulse response of a lowpass filter such as the moving average filter.

• Time-stretch, i.e. $y(t)=x(\alpha t)$, e.g. if you play an old $\mathrm{LP}-45$ vinyl disc at $33 \mathrm{rpm}$, the time-stretch coefficient would be $\alpha=33 / 45$

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3

(Difficulty: $\star$ ) The impulse response of a room can be recorded by producing a sharp noise (impulsive sound source) in a silent room, thereby capturing the scattering of the sound produced by the walls.
The impulse response $h[n]$ of Lausanne Cathedral was measured by Dokmanic et al. by recording the sound of balloons being popped ( hear it! ).

The balloon is popped at time $n=0$ and after a number of samples $N$, the reverberations die out, i.e. $h[n]=$ 0 for $n<0$ or="" $n="">N$
The acoustic of this large space can then be artificially recreated by convolving any audio recording with the impulse response, e.g. this cello recording becomes this.
What are the properties of $h[n] ?$ (tick all the correct answers)

• Anticausal
• FIR
• BIBO stable

4

(Difficulty: $\star$ ) Let

Compute $y[-1], y[0], y[1], y[2]$ and write the result as space-separated values. E.g.: If you find $y[-1]=-2$ $y[0]=-1, y[1]=0, y[2]=1,$ you should enter

5

(Difficulty: $\star \star)$ Consider the filter $h[n]=\delta[n]-\delta[n-1]$,the signal

and the output $y[n]=x[n] * h[n]$.
Compute $y[-1], y[0], y[1], y[2]$ and write the result as space-separated values. E.g.: If you find $y[-1]=-2$ $y[0]=-1, y[1]=0, y[2]=1,$ you should enter

6

(Difficulty: $\star \star \star)$ Which of the following filters are BIBO-stable?
Assume $N \in \mathbb{N}$ and $0<\omega_{c}<\pi$

• The ideal low pass filter with a cutoff frequency $\omega_{c}: H\left(e^{j \omega}\right)=\left\{\begin{array}{ll}1 & |\omega| \leq \omega_{c} \ 0 & \text { otherwise }\end{array}\right.$
• The moving average: $h[n]=\frac{\delta[n]+\delta[n-1]}{2}$
• The following smoothing filter: $h[n]=\sum_{k=0}^{\infty} \frac{1}{k+1} \delta[n-k]$
• The filter $h[n]=\sum_{k=0}^{N-1} \delta[n-k] \sin \left(2 \pi \frac{k}{N}\right)$

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$h[n]$绝对可和，所以BIBO

7

(Difficulty: $\star \star)$ Consider an LTI system $\mathcal{H}$. When the input to $\mathcal{H}$ is the following signal

then the output is

Assume now the input to $\mathcal{H}$ is the following signal

Which one of the following signals is the system’s output?

8

(Difficulty: $\star$ ) Consider the system shown below, consisting of a cosine modulator at frequency $\omega_{0}$ followed by an ideal bandpass filter $h[n]$ whose frequency response is also shown in the figure; assume that the input to the system is the signal $x[n],$ whose spectrum is shown below.

Determine the value of $\omega_{0} \in[0,2 \pi]$ that maximizes the energy of the output $y[n]$ when the input is $x[n]$
Remember that $\pi$ must be entered in the answer box as pi.

9

(Difficulty: $\star$ ) Consider a lowpass filter with the following frequency response.

What is the output $y[n]$ when the input to this filter is $x[n]=\cos \left(\frac{\pi}{5} n\right)+\sin \left(\frac{\pi}{4} n\right)+0.5 \cos \left(\frac{3 \pi}{4} n\right) ?$

10

(Difficulty: $\star$ ) Consider a filter with real-valued impulse response $h[n] .$ The filter is cascaded with another filter whose impulse response is $h^{\prime}[n]=h[-n],$ i.e. whose impulse response is the time-reversed version of $h[n]$

The cascade system can be seen as a single filter with impulse response $g[n]$
What is the phase of $G(e^{j \omega}) ?$

11

(Difficulty: $\star$ ) Let $x[n]=\cos \left(\frac{\pi}{2} n\right)$ and $h[n]=\frac{1}{5} \operatorname{sinc}\left(\frac{n}{5}\right) .$ Compute the convolution $y[n]=x[n] * h[n],$ and write the value of $y[5]$
Hint: First find the convolution result in the frequency domain.

12

(Difficulty: $\star \star)$ Consider the system below, where $H\left(e^{j \omega}\right)$ is an ideal lowpass filter with cutoff frequency $\omega_{c}=$ $\pi / 4$

Consider two input signals to the system:

• $x_{1}[n]$ is bandlimited to $[-\pi / 4, \pi / 4]$
• $x_{2}[n]$ is band-limited to $[-\pi,-3 \pi / 4] \cup[3 \pi / 4, \pi]$
• Which of the following statements is correct?

Which of the following statements is correct?

• $x_{2}[n]$ is not modified by the system while $x_{1}[n]$ is eliminated.
• Both $x_{1}[n]$ and $x_{2}[n]$ are not modified by the system.
• $x_{1}[n]$ is not modified by the system while $x_{2}[n]$ is eliminated.
• Both $x_{1}[n]$ and $x_{2}[n]$ are eliminated by the system.

$x_2$的频域属于

• $x_{2}[n]$ is not modified by the system while $x_{1}[n]$ is eliminated.

13

(Difficulty: \star) $x[n]$ and $y[n]$ are two square-summable signals in $\ell_{2}(\mathbb{Z}) ; X\left(e^{j \omega}\right)$ and $Y\left(e^{j \omega}\right)$ are their corresponding DTFTs.
We want to compute the value

in terms of $X\left(e^{j \omega}\right)$ and $Y\left(e^{j \omega}\right) .$ Select the correct expression among the choices below.

• $X\left(e^{j \omega}\right) * Y\left(e^{-j \omega}\right)$
• $\frac{1}{2 \pi} \int_{-\pi}^{\pi} X\left(e^{j \omega}\right) Y^{*}\left(e^{j \omega}\right) d \omega$
• $\frac{1}{2 \pi} X\left(e^{j \omega}\right) Y\left(e^{-j \omega}\right)$
• $\frac{1}{2 \pi} \int_{-\pi}^{\pi} X\left(e^{j \omega}\right) Y\left(e^{-j \omega}\right) d \omega$
• $\frac{1}{2 \pi} X\left(e^{j \omega}\right) Y^{*}\left(e^{j \omega}\right)$
• $X\left(e^{j \omega}\right) Y\left(e^{-j \omega}\right)$

14

(Difficulty: $\star \star \star) h[n]$ is the impulse response of an ideal lowpass filter with cutoff frequency $\omega_{c}<\frac{\pi}{2} .$ Select the correct description for the system represented in the following figure?

Hint: Use the trigonometric identity $\cos (x)^{2}=\frac{1}{2}(1+\cos (2 x))$

• A highpass filter with gain $\frac{1}{2}$ and cutoff frequency $\pi-\omega_{c}$
• A lowpass filter with gain 1 and cutoff frequency $\omega_{c}$
• A highpass filter with gain $\frac{1}{4}$ and cutoff frequency $\omega_{c}$
• A lowpass filter with gain 1 and cutoff frequency $\omega_{c} / 2$
• A highpass filter with gain 1 and pass band $\left[\omega_{c}, \pi-\omega_{c}\right]$
• A lowpass filter with gain $\frac{1}{2}$ and cutoff frequency $2 \omega_{c}$

• A highpass filter with gain $\frac{1}{2}$ and cutoff frequency $\pi-\omega_{c}$

15

(Difficulty: $\star \star \star)$ Consider the following system, where $H\left(e^{j \omega}\right)$ is a half-band filter, i.e. an ideal lowpass with cutoff frequency $\omega_{c}=\pi / 2$

Assume the input to the system is $x[n]=\delta[n]$. Compute
$\sum_{n=-\infty}^{\infty} y[n]$
Hint: Perform the derivations in the frequency domain.