# Tag Info

14

Tips for DSP self-study huh. Well, ...studying 'signals and systems' is a great idea and having Matlab software means you have the tools to learn an awful lot. I think Dr. Steven Smith's book "The Scientist and Engineer's Guide to Digital Signal Processing", which you can read online for free, is a terrific source of fundamental DSP information. Dr. Smith is ...

10

Oppenheim and Willsky's Signals and Systems or Lathi's Linear Systems and Signals are intended for Sophomores who have only a single semester of differential equations under their belts, so it is a bit unfair to criticize them for leaving out the functional analysis and the conformal mapping. At the sophomore level my favorite book is Siebert's Circuits, ...

8

You can visit the MIT OpenCourseWare. A set of 20 video lectures by professor Alan V. Oppenheim.

8

You seem to have a number of misunderstandings, which I'll try to clarify while also trying to help with your questions. The entropy of a source $H(S)$ gives the average codeword length to encode a given source alphabet. i.e. it is the average number of bits per symbol required to encode the information in the source. While this is true, I think it's not ...

7

Get, Read and Solve the following books: Signals and Systems. Discrete Time Signal Processing. Digital Signal Processing. Take the following courses: Coursera - Digital Signal Processing. edX - Discrete Time Signal Processing. edX - Signals and Systems: Part I, Part II. edx - Discrete Time Signals and Systems: Part 1: Time Domain, Part 2: Frequency Domain....

6

Lets say we want to transmit a sequence of discrete data $\left\lbrace x[n] \right\rbrace$. But because we are living in analog world, the sequence must be modulated. Call $T_s$ is symbol duration and use a set of orthonormal waveforms $\left\lbrace p_n(t) = p(t-nT_s), n \in \mathbb{Z} \right\rbrace$, (baseband) signal $x(t)$ can be written as \begin{...

6

Here I expected $y(n)$ is to be computed by convolving $x(n)$ with $h(n)$, but in the equation given by Wikipedia it is shown as a matrix multiplication $y(n) = h^H(n).x(n)$. Are these two operations(convolution and matrix multiplication) same here?. The system is an FIR system, so the vector multiplication here is equivalent to convolution --- for ...

5

You should take a look at "Mathematical Methods and Algorithms for Signal Processing" by Moon and Stirling. The only downside is its long list of errata, so hopefully there will be a new edition soon. A beautiful book about the Fourier transform as it's used in signal and system theory is "The Fourier Integral and its Applications" by Papoulis. It's quite ...

5

The reason why almost all linear adaptive equalizers are implemented as FIR filters is that FIR filters are always stable and that there exist relatively simple and effective adaptation algorithms. Note that much work has been done on adaptive IIR filters (e.g., this book by Phillip Regalia), but in practice FIR filters are still the preferred option. Note ...

5

If you have two DFTs $A[k]$ and $B[k]$ (note the correct representation of a sinusoid at DFT bin number $1$) A = [0,-j,0,0,j]; B = [1,1,1,1,1]; with the corresponding time-domain sequences $a[n]$ and $b[n]$ a = ifft(A); % [0, 0.38042, 0.23511, -0.23511, -0.38042]; b = ifft(B); % [1,0,0,0,0]; then the multiplication of the time-domain sequences $c[n]... 5 There are, a few discrepancies that might be making a difference here. My suggestion would be to edit the question for clarity. There are quite a few assumptions that lead to non-straightforward thinking about the problem which I have tried to address to an extent and I would be happy to modify the response in light of more information. In machine ... 5 Complex channel coefficient is just a way to represent the independent real coefficients. You just need to generate h = [h_0, h_1, h_2] = [hR_0, hR_1, hR_2] + 1i * [hI_0, hI_1, hI_2]. The independence/correlation between coefficients depend on your model. And if I were not wrong, the number of element of h is the order of your MA model. The idea behind ... 4 The DSP neophyte who has some mathematical maturity may want to start with Martin Vetterli, Jelena Kovačević, Vivek Goyal, Foundations of Signal Processing, 2014. which is freely available online. The authors have also made their two other books freely available online: Jelena Kovačević, Vivek Goyal, Martin Vetterli, Fourier and Wavelet Signal Processing, ... 4 Ok, there is some misconceptions in your question. I strongly recommend you to read a little more about the topics, but I will try to help you a little. My answers and some comments: ...linear equalizer is a filter that can undo these channel effects. When the channel coefficients w are unknown, we perform blind equalization. In this scenario, we ... 4 The book doesn't say that the impulse response must be zero for an ideal channel. It says that an ideal channel has exactly one, and not more than one, non-zero component, i.e. the ideal channel's impulse response is an impulse, which means that the signal is only delayed but not distorted. 4 "Digital Signal Processing: A Computer-Based Approach" by Sanjit Mitra is what you need I guess, especially the exercises at the end of each chapter. There is a booklet on the Internet again by Mitra, named Digital Signal Processing Laboratory Using MATLAB. The other option could be Practical Signals Theory with MATLAB Applications. 4 (1) I'm not sure my intuition is right Almost, but you're missing the fact that if you're sampling at 4000kHz, the signals alias; after sampling at 4000kHz, a sine wave at 1500Hz is indistinguishable from one at 2500Hz or 5500Hz, etc. Also, because the work is being done in complex numbers, negative frequencies have meaning:$e^{-j \omega t}is different ... 4 Here's the pure math: $$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{i \frac{2\pi}{N} nk }$$ $$x[n] = \frac{1}{8} \left[ (5.657 + i 5.657) e^{i \frac{2\pi}{8} n 3 } +(5.657 - i 5.657) e^{i \frac{2\pi}{8} n 5 } \right]$$ $$e^{i \frac{2\pi}{8} n 5 } = e^{-i \frac{2\pi}{8} n 3 }$$ \begin{aligned} x[n] &= \frac{1}{8} \left[ (5.657 + i 5.657) ... 3 I would let them read the paper about the Non Local Means Filter: Antoni Buades, Bartomeu Coll, Jean Michel Morel - On image Denoising Methods. The paper is readable and it is a great introductory to the Denoising operation in the context of Image Processing. Also the Non Local Means is a very decent method (Result wise) even in our time. 3 I don't understand the subscript n notation, however, in the least squares problem that is given by: \begin{equation} {\bf{y}}={\bf{H}}{\theta}+\bf{n}, \end{equation} where {\bf{n}}\sim\mathcal{N}(\bf{0}, \sigma^2I_N) is a zero mean additive white Gaussian noise and I_N is the N \times N identity matrix, the maximum likelihood and the least squares ... 3 Paul R is right: your system in not time-invariant, so the impulse response doesn't mean much. This could actually be a typing error o your site, more common is y(n) = k*y(n-1)+x(n). This being said, you can still calculate impulse response my simply starting at n = 0 and evaluate the difference equation one step at a time. For 1,2,3,4 ... so the impulse ... 3 I would add to the list the book "Digital Filters", by Richard Hamming. A short classic, rather than a heavy tome. 3 The fundamental idea to keep in mind is that in a wireless channels with reflections, if you transmit s(t), you'll receiver(t)=\sum_{i=1}^Na_is(t-\tau_i).$$Another important idea is that whether the channel is flat or not depends only on s(t). For instance, let's say that the symbol time is T_s. Let's call the longest delay in the channel \tau_{... 3 For question 1: apply the definition of time invariant: find the output as normal; find the output with the same input but delayed by T$$ y_1(t) = \frac{dx(t)}{dt}\\ y_2(t) = \frac{dx(t-T)}{dt}\\ $$Does y_2(t) = y_1(t-T) ? For question 2: Find a definition of stability and apply it. For example, for a system to be BIBO stable it needs to have$$ \... 3 Question 1: Yes, it is a multipath Rayleigh Channel with 1000 $taps. You need to know that when you generate complex coefficients, and the real part and the imaginary part of this coefficients are randomly chosen according to a Gaussian distribution, the magnitude will have a Rayleigh distribution and the phase will have a uniform distribution between$ ...

3

From the definition of the process you know that $$x_{n+1}=x_n-0.2x_{n-1}+w_{n+1}\tag{1}$$ Since $w_n$ is white you can't predict it, so the best linear predictor for the given process is the filter $$P(z)=1-0.2z^{-1}\tag{2}$$ which is a first order filter. It estimates the future sample $x_{n+1}$ by computing $$\hat{x}_{n+1}=x_n-0.2x_{n-1}\tag{3}$$ ...

3

To build on Laurent's answer, here is an example. Top frame shows an example transient signal: a damped sine wave. As the signal decays very quickly, the first 0.1s of the signal is the most interesting. Second frame shows hann window. Hann window is almost zero over that first 0.1s. Third frame shows what happens when you apply hann window to the ...

3

An M-QAM modulation contains M different constellation symbols. Hence, in one symbol you can encode $\mu=\log_2(M)$ bits. (i.e. 6 bits for 64-QAM for example). Now, the bits are equally distributed along the real and the imaginary part of the constellation, i.e. you have $\frac{\mu}{2}$ bits for the real and imaginary part. To encode $\frac{\mu}{2}$ bits, ...

3

A PRN sequence is a Pseudo-Random Noise sequence, often generated by using an Linear Feedback Shift Register (LFSR) with the feedback taps done by using a primitive irreducible polynomial in GF{2}, which is the Golois Field of 2 elements. When a primitive and irreducible polynomial in GF{2} is used, the LFSR will produce a "maximum length sequence", meaning ...

3

Let me try to establish the relation between the univariate PDF for a real Gaussian and the univariate PDF for a complex proper (i.e. circular symmetric) Gaussian. You know that $p_x(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp(\frac{-x^2}{2\sigma^2})$ is the PDF of a real-valued Gaussian with variance $\sigma^2$. We write $x\sim\mathcal{N}(0,\sigma^2)$ to denote ...

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