Questions tagged [homework]

Homework means the asker is requesting help with school homework. This lets potential answerers know that they should guide the student in solving the problem, rather than simply showing the complete answer.

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195 views

Finding the Period of a Signal by Its DTFT Data

find period of DTFTs where vertical lines represent Dirac deltas
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1answer
44 views

Properties of DTFT to Infer the Inverse DTFT of Altered Data [closed]

Given a signal $ x \left[ n \right] $ and its DTFT $ X \left( {e}^{j \omega} \right) $. Which property of the DTFT allows you to easily compute the inverse DTFT of $ \frac{4 X \left( {e}^{j \omega} \...
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1answer
594 views

Calculate the Inverse DTFT of the DTFT Derivative in Terms of $ x \left[ n \right] $

Consider the signal $ x \left[ n \right] $ and its DTFT transform $ X \left( {e}^{j \omega} \right) $. Assume $ X \left( {e}^{j \omega} \right) $ is differentiable. What is the Inverse DTFT of: $$ j ...
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1answer
116 views

How to evaluate CCDE at a given frequency?

Consider the following Constant Coefficient Difference Equation (CCDE) given below $$y[n] - \frac{1}{2}y[n-1] = 2x[n] - 5x[n-1] - x[n-2]$$ Let $H(e^{j\omega})$ denote the transfer function of this ...
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1answer
51 views

Meaning of a formula about transient frequency bins

I'm trying to understand the following formula in this paper: Sadly, the paper does not define most of the symbols it is using, so my guesses so far are: $T_{tr}$ refers to the sampling interval i....
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1answer
33 views

Signal analysis using a buffer [closed]

I am performing an analysis of an output signal in the interval $t<0$ from the operational amplifier given in the following image where $v_{C_1}(0^-)$ and $v_{C_2}(0^-)$ are the initial conditions ...
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1answer
841 views

Determine if $ y[n] = ny[n-1] + x[n]$ is linear time invariant and BIBO stable

Check if the following system is linear time invariant and BIBO stable.. $$ y[n] = ny[n-1] + x[n] $$ for $n\ge 0$. We are also given that the system is at rest (i.e. $y[−1] = 0$). I know that to ...
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1answer
714 views

Ratio of the power in the sidebands to the total power in conventional AM

Message signal and carrier signal are given as $m=\cos(2 \pi \cdot f_{m} \cdot t) $ $c=A_{c}\cdot \cos(2 \pi \cdot f_{c} \cdot t) $ I was asked to find modulation index $a$ so that ratio of the ...
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1answer
41 views

Need help deriving the energy of a signal

I am solving old exam problems in preparation of my exam in signals, and I'm having trouble with a question. For a continuous aperiodic signal with the spectrum $X(\omega)=\exp(-\omega^2)$ I am to ...
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1answer
70 views

Homework Help: What does $h[0] = 1$ represent? What is $\ln \big| H(e^{j \omega})\big|$?

I have been staring at this problem for a week now... Suppose $H(e^{j \omega})$ is the frequency response of a stable and causal minimum-phase discrete-time system with $h[0]=1$ ($h[n]$ is the ...
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1answer
257 views

Pole zero plot, normalizing frequency response plot?

I'm asked to plot the frequency response (amplitude) given a specific pole-zero diagram. $$ H(z) = H_0 \frac{\prod\limits_{m=1}^{M} (z - q_m)}{\prod\limits_{m=1}^{M} (z - p_m)}$$ $$ H(e^{i\omega}) =...
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1answer
92 views

Is this two input discrete system linear? [closed]

Given two inputs $\: x_1[n]\: x_2[n]\:$Is the system $\:y[n]=x_1[n]\times x_2[n]\:$ linear ? My Approach: $(x_1\times x_2)[n]=S_1[n]\rightarrow Y_1[n]$ $(x_3\times x_4)[n]=S_2[n]\rightarrow Y_2[n]$ $...
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3answers
592 views

Distinguishing FIR and IIR from difference equation

Find the transfer function of the difference equation $$y_n = x_n + 1.2y_{n-1}$$ I fail to understand how one can distinguish between an FIR filter and an IIR filter by looking at the equation given ...
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1answer
6k views

How to find the phase spectrum of a rectangular pulse? (Fourier Transform)

How to find the phase spectrum of a rectangular pulse? The Fourier transform of a rectangular pulse $$ x(t) = \begin{cases} 1, & \text{for $|t| \le \tau /2$ } \\ 0, & \text{otherwise} \end{...
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2answers
308 views

Interpolation and decimation

I have a sinusoid in continuous time, with a frequency of 18kHz, it is sampled ideally with a continuous to discrete convertor, with a frequency of 27kHz. After that, we change the sampling speed in ...
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1answer
48 views

impulse response of a system from its step response;confusion

If the step response of a system is given as $$c(t)=(10e^{-5t}-2e^{-10t}+1)u(t)$$ then its impulse response is _______? My Approach: we know: $$h(t)= \frac {d}{dt} c(t)$$ Process(1): $$h(t)= \{ ...
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1answer
49 views

Clarification on Discrete-time convolution and signal shifting

What are $y_1[n]=x_1[n]*h[n]$ and $y_2[n]=x_2[n]*h[n]$? $$x_1[n]=(0.1)^nu[n],\quad x_2[n]=(0.2)^n,\quad h[n]=(0.3)^nu[n]$$ I am confused about how to calculate $y_{1}[n]$. The formula for discrete ...
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1answer
344 views

How to sketch the following discrete time signal $\sum_{k=0}^{n} \delta[n-k]$?

I tried to solve it like in the picture but It feels so wrong though.
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1answer
75 views

Fourier Coefficients

Consider the signal $$x(t)=\cos(2\pi t)$$ Since $x(t)$ is periodic with a fundamental period of $1$, it is also periodic with a period of $N$, where $N$ is any positive integer. What are the ...
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1answer
351 views

Step response given impulse response $h = e^{-|t|}$

I was given the above homework problem and I know how the step response is achieved, but I'm not sure how to set up the integrals in order to evaluate the problem. I've attached my work, but it is ...
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1answer
77 views

Fourier components of $\cos(2\pi f_1t)$

I have the signal $s(t) = \cos(2\pi f_1t)$ and I am looking for its components vs the Fourier basis, over the interval $[0, T]$. The formula for computing the coefficients is $$ s_n = \int_{t_0}^{t_1} ...
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1answer
43 views

Pulse wave question

Wikipedia, fount of all knowledge (Ha! LOL), gives a formula for a pulse wave here: The formula is: $$f(t)=\frac{\tau}{T}+\sum_{n=1}^{\infty}\frac{2}{n\pi}\sin\left(\frac{\pi n \tau}{T}\right)\cos\...
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1answer
56 views

Proof of $E[y(n)]=E[x(n)] \, \sum h(k)$

If $h(n)$ is the impulse response of the discrete LTI system, $x(n)$ is the white noise process with variance $\sigma_x^2$ and $y(n)$ is the output, prove that $$\mu_y = \mu_x \, \sum_{k=0}^\...
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0answers
156 views

Equation for impulse train

I am looking for a formula (Fourier series) to generate an impulse train waveform - a spike-wave with amplitude and period both $1$ – so that $f(x)$ has value $1$ at $x = 1,2,3,4...$ and $f(x)$ has ...
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1answer
71 views

Time-variance and causality of $y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$

Is the following system time invariant and/or causal? $$y(t)=\sum_{k=-\infty}^{+\infty}x(t)\delta(t-kT)$$ I think that it isn't time invariant and it is causal, and my thought is based on the ...
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1answer
186 views

How do I find the ROC of a system if it has no poles

The output of a system of discrete time $y[n]$ is corellated with the input $x[n]$ through the equation $y[n]$. $$y[n] = \frac 13\big(x[n-1]+x[n]+x[n+1]\big)$$ It then asks me to find the system ...
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1answer
119 views

Time-invariance, causality and stability of $h(t)$ of four given systems

Question: The impulse response functions of four linear systems $S_1,\ S_2,\ S_3,\ S_4$ are given respectively by \begin{align} h_1(t)&=1\\ h_2(t)&=u(t)\\ h_3(t)&=\frac{u(t)}{(t+1)}...
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1answer
118 views

Find Fourier Transform of signal that satisfies this equation

I have attempted it this way: $G(j\omega)$ is a rectangular pulse in frequency domain. Using results, inverse Fourier transform of $G(\omega)$ will be $$g(t)=\frac{\sin(2t)}{\pi t}$$ Now given that $...
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1answer
557 views

How to find inverse of convolution integral?

If $x^{-1}(t)$ and $ y^{-1}(t)$ denote the integrals of x(t) and y(t) defined by $x^{-1}(t)=\displaystyle\int_{-\infty}^{t} x(\lambda)d\lambda$ $y^{-1}(t)=\displaystyle\int_{-\infty}^{t}y(\lambda)d\...
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2answers
2k views

how to evaluate derivative of convolution integral?

If the signal $x(t)$ has ordinary first derivative $\dot x(t)$, then $\dfrac{d}{dt}\big(x(t)\star y(t)\big)$ is: (a) $\dot x(t)y(t)$ (b) $x(t)\dot y(t)$ (c) $\dot x(t)\star y(t)$ (d) $\...
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1answer
664 views

Time shifting and time reversal

For the problem given in the fig. how to solve it? My approach: Let $y(n)$ be the first output (e.g for 1st case output of system $A$ is $y_1(n)$; for 2nd case output of system $B$ is $y_2(n)$, then ...
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1answer
126 views

Drawing the modulus from a Transfer function

As you may guess from my other questions i am a student so pardon me any ignorance and guide me to the truth. Thanks In this exercise from an exam i was given that transfer function with poles and ...
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1answer
32 views

Evaluating different inputs into a given system

We need to find which signal wont be the same after going through this sytem. My approach is this one: I took the filters frequency and did 2pi * 1/4 and gave me this "limit" 1.5707 (2pi because ...
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2answers
95 views

Finding the correct $x[n]$ from a given DFT $x[k]$ [closed]

I'm struggling to solve this question from an exam. The question gives you a set of IDFTs from a given DFT, only one of those is the correct. The first thing I did is notice that $2\sin(\pi k)$ will ...
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1answer
811 views

proof of Autocorrelation property of DFT

I am facing problem in proving the Auto-correlation property of Discrete Fourier Transform (DFT), that is $$\mathcal{DFT}\left\{\sum_{r=0}^{N-1}x[r]x^*[r+n]\right\} = X[k]X^*[k]= |X[k]|^2$$ where $X[...
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2answers
380 views

Inverse system of a system with integral

I am trying to find the inverse system of the following (I tried finding the mathematical inverse function but since it is not the same I am not so sure) . Can someone help me find it? $$ y(t)=\int_{...
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1answer
36 views

Convolution including $\delta(t-5)$

I know the two properties of convolution that are related to my question $\quad x(t)*\delta(t)=x(t) $ $\quad x(t)*\delta(t-t_0)=x(t-t_0)$ But my question is, how can I use those two to calculate $$ ...
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1answer
262 views

Cross-correlation - Why is there a complex part after FFT/IFFT when deleting the mirrored part of the spectrum?

I want to detect a special shape in my time series and I apply an matched filter using cross-correlation to increase the SNR. ...
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3answers
102 views

different moving average filters differences

Suppose we want to use a moving average filter (3-point moving average) for $X[n]$ (where $Y[n]$ is output). What are the differences among $Y[n]=\frac13X[n-1]+\frac13X[n]+\frac13X[n+1]$, $Y[n]=\...
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1answer
527 views

Impulse response function from pole-zero graph

I know that that the poles of this system are: $s_1 = -1 + 2j$ and $s_2 = -1 - 2j$. I think to solve this problem I would need to find the laplace of this plot, with the formula: $$ W(s) = \frac{\rm ...
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2answers
126 views

How to find $h[n]$ system response of this equation?

$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response. I transformed this system via using Z transform: $$\frac{Y(z)}{X(z)}=H(z)=\frac{z^4 - 10z^...
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1answer
53 views

Find the coefficient of an AR-model

How can I infer the coefficients for an AR(2) model given an autocorrelation plot? What I have tried so far: I can see this a cos-wave of 0.4 Hz, and then i find the values for each step $(1, 0.9211, ...
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1answer
222 views

Autocorrelation and PSD

Let $X(t)$ and $Y(t)$ be two orthogonal processes with power spectral densities $$S_{xx}(f) = S_{yy}(f)=\begin{cases} 1-\lvert f\rvert, & \lvert f\rvert<1 \\[1ex] 0,& \text{otherwise} \end{...
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1answer
2k views

The error probability of 16QAM

The signal constellation for a communication system with 16 equiprobable symbols is shown as below. The channel is AWGN with noise power spectral density of $N_ 0/2.$Using the union bound, find a bound ...
2
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1answer
54 views

Transmitting over Unknown Fading Channel

I have a system with no diversity and the channel's coefficients change every $T = 10\text{ bits}$ transmitted. The channel estimation system has a flaw and detects the channel's coefficients equal to ...
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2answers
140 views

Problem identifying the analytic expression of such determined signal

I came across this problem I am supposed to find the Fourier transform of $g(t)$, but I am not able to find the analytical expression of such signal. The teacher suggests that I should consider ...
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1answer
216 views

Reconstruction of a signal using an ideal DAC

I think I was able to do part (a) correctly, but now am stuck on part (b). I don't know how to approach this part or what is the theory behind it. My MATLAB code is shown below. Please let me know if ...
2
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1answer
617 views

Relating transfer functions with step responses

Relate the transfer function to its' corresponding step response. First, I tried setting up the poles and zeros of the transfer functions. This helped a bit since I know that $G_A (s)$, $G_B(s)$ and $...
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1answer
180 views

Smallest sampling frequency to fully reconstruct a signal

Given the spectrum of analog signal $x_a(t)$ which is imaginary and band-limited, find the lowest sampling frequency to be able to reconstruct $x_a(t)$ from samples $x[n]$. My attempt: Bandwidth of ...
4
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2answers
101 views

Filtering without using any transforms

A composite signal is defined as $$x(t) = \sin(2\pi 10t) + \sin(2\pi 40t) + \sin(2\pi 60t) + \sin(2\pi 50t)$$ Separate the four frequencies without using any of the transform techniques.