• ##[[Signal Energy and Power_(ECE301Summer2008asan)|Signal Energy and Power]] ##[[Continuous-Time and Discrete-Time_(ECE301Summer2008asan)|Continuous-Time and Discrete-Time]]
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  • (a) The FT of <math>X(j\omega)</math> of a continuous-time signal x(t) is periodic (b) The FT of <math>X(e^{j\omega})</math> of a continuous-time signal x[n] is periodic
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  • .../math> of the signal x[n] is also periodic with period N. For the periodic signal x[n], find the values of <math>a_0,a_1,...,a_{N-1}.</math> Express your an 1)b)Evaluate the value of <math>(1/N)*\sum_{n=<N>}|x[n]|^2</math> for the signal x[n] given in part (a).
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  • The command is ifft. It takes in a vector representing your signal and produces a vector of the fourier series coefficients. Two examples are The signal is represented by the graph below and is periodic for all time:
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  • ...he Fourier Transform.... In particular, Fourier reasoned that an aperiodic signal can be viewed as a periodic singal with an infinite period." An example of ...nd the sound of the musical chord represented by these notes (the function/signal itself).
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  • ...ecause it does not specify from what the signal is being recreated. If the signal is for example not band-limited, it cannot be reconstructed at all. ...an the Nyquist rate in rare cases you are able to properly reconstruct the signal.
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  • [[Category:signal processing]] 2) Digital Signal = a signal that can be represented by a sequence of 0's and 1's.
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  • *<math>\omega_m</math>: Maximum frequency in a band-limited signal (<math> = max(\{|w|\ :\ w \neq 0\})</math> ...hen the band-limited signal can be uniquely reconstructed from the sampled signal.
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  • [[Category:signal processing]] <li>Signal Characteristics</li>
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  • [[Category:signal processing]] <strong>Continuous-time:</strong> (a.k.a. Dirac delta function)<br/>
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  • == Continuous-time domain == *[[CT Time-averaged Power of a Signal over an infinite interval_ECE301Fall2008mboutin]] {{:CT Power of a Signal_
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  • ...Fourier transform of x[n], which is the sampled signal of continuous time signal x(t) <br>
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  • [[Category:signal processing]]
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  • [[Category:signal processing]]
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  • Note: PM refers to the official course book, Digital Signal Processing, 3rd edition, J.G. Proakis and D.G. Manolakis. ...due.edu/~bouman/ece438/lecture/module_1/1.1_signals/1.1.1_signal_types.pdf Signal Types]
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  • == Continuous Signal == Continuous signal is a signal that varies with time, and can be represented as a function of time, x(t).
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  • == Periodic Signal == Notice, the signal is the same throughout each cycle.
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  • A discrete time signal is periodic if there exists T > 0 such that x(t + T) = x(t) A continuous time signal is periodic if there exists some integer N > 0 such that x[n + N] = x[n]
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  • A continuous time signal is periodic if there exists a value <math> T </math> such that <math> x(t + A discrete time signal is periodic if there exists a value <math> N </math> such that <math> X[n +
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  • A continuous time signal x(t) is periodic if there exists T such that x(t + T) = x(t) for all t. <br A discrete time signal x[n] is periodic if there exists some integer N such that x[n + N] = x[n] f
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  • A Continuous Time signal is said to be periodic if there exists <math>\ T > 0</math> such that <math A Discrete Time signal is said to be periodic if there exists <math>\ N > 0</math> (where N is an
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  • == Periodic Signal Definition == *For a Continuous-time signal
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  • For a continuous-time signal <br> ...m_{T \to \infty} {\frac{E(\infty)}{2T}} = 0 ................ Finite-energy Signal</math><br>
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  • Computation of Signal Energy and power. Source for definition Of Continuous Signal: Wikipedia.
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  • == For a Continuous Time Signal==
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  • A periodic signal is one that for a given real number "a": ===Periodic Signal===
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  • A signal is periodic if there exists some T>0 such that: A signal is NOT periodic if the converse is true, there DOESN'T exists some T>0 such
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  • ==Periodic Signal== A continuous time (CT) signal is periodic if it there exists some T such that x(t+T)=x(t) for all t.
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  • ==Periodic Signal== to prove a CT signal is continuous we must prove that there exists a value T such that x(t) = x(
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  • '''Changing a Periodic Continuous Time Signal to a Non-Periodic Discrete Time Signal''' ...nsider the continuous time signal <math>x(t)=sin(t)</math>. Plotting this signal yields a smooth waveform that repeats itself with period <math>T=2\pi</math
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  • == Continuous to discrete time signal== I used the signal <math>y = cos(n)\,</math> as the signal of my graph
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  • === Periodic Continuous Time Signal === ...y people used in Homework 1 for their example of a periodic function. The signal repeats itself at intervals of <math> 2\pi </math>.
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  • == CT periodic signal == An example of a periodic signal in continuous time is:
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  • I chose to use the CT (continuous time)periodic signal: y(t) = cos(t). The signal can be expressed as both periodic and non-periodic in DT (discrete time).
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  • A system is called time invariant if for any input signal x(t)(x[n]) and for any t0 belongs to R, the response to the shifted inputX( ...= 10 x(t-t0)where as a system is called time variant when we find an input signal for which the condition of time invariance is violated.
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  • ...ework 1 were boring (including mine) so I thought I'd broaden the periodic signal pool. I chose the CT signal: <math>x(t) = |2*cos(.5*t)|</math> . A graph of this signal in continuous time is shown below.
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  • <b>Changing a Periodic Continuous Time Signal to a Non-Periodic Discrete Time Signal</b> The signal I chose for this part can be found [[HW1.4 Wei Jian Chan - Periodic and Non
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  • 1.This is a sine function of period 2. Function is sin(pi*t). Continuous Signal. 2. '''Periodic DT Signal'''This is the discrete signal of the same function in 1 with sampling time of 0.075. I got the diagram on
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  • A system is called "'''time invariant'''" if for any input signal x(t) in continuous time or x[n] in discrete time and for any time <math>t_0 A system is called "'''time variant'''" if for any input signal x(t) in continuous time or x[n] in discrete time and for any time <math>t_0
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  • == Example of Computation of Fourier series of a CT SIGNAL == ==The Signal==
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  • ==Response to a Signal from Question 1== I will use my signal from Question 1.
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  • == Example of Computation of Fourier series of a CT SIGNAL == The function y(t) in this example is the periodic continuous-time signal cos(t) such that
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  • A continuous-time Linear Time-Invariant (LTI) system defined for the purpose of this page wil where v(t) is an input signal dependent on the parameter of time.
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  • == Example of Computation of Fourier transform of a CT SIGNAL == Let the signal x(t) be equal to:
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  • Let the signal <math>X(\omega)</math> be equal to: The Inverse Fourier Transform of a signal in Continuous Time is:
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  • The response to the input signal <math>z^n</math> is <math>H(z)z^n</math>, giving
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  • '''Question:''' Compute the Fourier transform of the signal x(t) equal to: The Fourier Transform of a signal in Continuous Time is defined by:
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  • ...Fourier transform exists if the signal is absolutely integrable or if the signal has a finite number of discontinuities within any finite interval. (See Pag :This is useful for signals that fail to satisfy the previous properties of a signal that is guaranteed a Fourier Transform.
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  • What is it? Sampling is a process of measuring a CT signal x(t) at some specific values of time t. ...ample a continuous time signal x(t) at point t-1, t-2 and t-3. The sampled signal can represented by the formula <math>y[n] = x(nT)\,</math>
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  • ==Reconstructing a signal from its samples using Interpolation== ...an important procedure we know as interpolation we can obtain the original signal of the function.
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  • ...ved by the use of a periodic impulse train multiplied by a continuous time signal, <math>x(t)</math>. The periodic impulse train, <math>p(t)</math> is refer
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  • ...continuous-time signal to discrete-time signal, process the discrete-time signal using a discrete-time system and convert it back to continuous time. ...and X(j<math>\omega\,</math>) be the continuous Fourier transform of that signal. Then,
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  • ...the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate. ...signal and <math>X(W)\,</math> be the continuous Fourier transform of that signal
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  • Let <math>x(t)</math> be a signal with <math>X(\omega) = 0</math> when <math>|\omega| > \omega_m</math>. <math>\omega_m</math> Maximum frequencye for a band limited signal
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  • ...the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate. ...signal and <math>X(w)\,</math> be the continuous Fourier transform of that signal (which exists if <math>x(t)\,</math> is square-integrable)
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  • ...AB. By use of sampling a continuous signal can be converted to a discrete signal, manipulated via a computer program and then converted back into a continuo
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  • ...u have used a good sampling rate, you should be able to reconstruct the CT signal without much fuss.
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  • ...ation is defined by the authors of our book as the fitting of a continuous signal to a set of sample values, and is a commonly used procedure to reconstruct ...tion of the CT signal from the sampled signal approximates the original CT signal better.
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  • Let x(t) be a signal with <math>\chi(\omega)=0</math> when <math>|\omega|<\omega_m</math>. P: A real-valued signal x(t) is known to be uniquely determined by its samples when the sampling fr
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  • ...actly, from samples. More so, interpolation is the fitting of a continuous signal to a set of sample values. Interpolation can also be defined as a specific
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  • The continuous-time Fourier transform provides us with a representation for signals as linear c ...the complex variable s, it is referred to as the Laplace transform of the signal. The complex variable zs can be written as <math>s=\sigma+j\omega</math>, w
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  • ...rm is used on continuous signal while z transform is used for the discrete signal. The z- transform of a general discrete signal x[n] is defined as
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  • #'''Signal Reconstruction Using Interpolation:''' the fitting of a continuous signal to a set of sample values ...nals (CD to MP3 albeit a complicated sampling algorithm, MP3 is less dense signal)
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  • :(c) an ability to determine the response of linear systems to any input signal convolution in the time domain. [1,2,4;a,e,k] :(e) an ability to determine the response of linear systems to any input signal by transformation to the frequency domain, multiplication, and inverse tran
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  • #'''Signal Reconstruction Using Interpolation:''' the fitting of a continuous signal to a set of sample values ...nals (CD to MP3 albeit a complicated sampling algorithm, MP3 is less dense signal)
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  • ##[[Signal Energy and Power_Old Kiwi]] ##[[Continuous-Time and Discrete-Time_Old Kiwi]]
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  • (a) The FT of <math>X(j\omega)</math> of a continuous-time signal x(t) is periodic (b) The FT of <math>X(e^{j\omega})</math> of a continuous-time signal x[n] is periodic
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  • .../math> of the signal x[n] is also periodic with period N. For the periodic signal x[n], find the values of <math>a_0,a_1,...,a_{N-1}.</math> Express your an 1)b)Evaluate the value of <math>(1/N)*\sum_{n=<N>}|x[n]|^2</math> for the signal x[n] given in part (a).
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  • Note: PM refers to the official course book, Digital Signal Processing, 3rd edition, J.G. Proakis and D.G. Manolakis. * Basic Signals and Signal Properties
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  • * Signal properties (even/odd, periodicity, power, energy, etc.)
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  • ...thcal{F}</math> of {a*x(t)+b*y(t)} will be equal to {a*X(w)+b*Y(w)} if the signal is truly linear. Signal <math>x(t)=1, 0<t\le1; 2, 1<t\le2; 0, else=u(t)+u(t-1)-2u(t-2)</math>
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  • '''== Time Shifting Property of Continuous-Time Fourier Series ==''' <br> When a time shift is applied to a periodic signal x(t), the period T of the signal is preserved.<br>
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  • ...the DT Fourier transform look like if the discretization represents the CT signal well?". Should we organize another recitation on that topic?--[[User:Mbou
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  • [[Category:signal processing]] ...frequency spectrum of a signal the faster we sample it. Naturally, if the signal changes much faster than the sampling rate, these changes will not be captu
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  • '''Q:''' What is a digital signal? '''A:''' A signal that can be represented by a sequence of 0's and 1's.
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  • ...tions caused by analog circuitry. One area that this can be applied is in signal reconstruction, where a low pass analog filter is used on the output of a d ...ed to relax requirements on analog low pass filter design while decreasing signal distortion.
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  • ...s time signal (consisting of infinite number of points) to a discrete time signal (finite points). This process enables the conversion of analog signals to ...period T). This can be achieved by multiplying the given continuous time signal by a train of dirac delta functions separated by the time period T. This c
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  • xc(t)=continuous time signal x[n]=discrete time signal
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  • ...ermines the number of pixels the camera uses to represent the "continuous" signal (e.g. a mountain, or your smiling significant other) that your digital came Thus the digital camera '''''samples''''' the continuous signal, with a period <math>T</math> (shutter speed) and "on" for length <math>tau
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  • ...e the magnitude of each frequency component's contribution to the original signal. Finally, the Fourier Transform is calculated to express these coefficients ...at is commonly referred to as the "spectrum" of the original discrete-time signal, x[n]. To demonstrate why this is the case, consider the following discrete
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  • ...s attenuating high frequency portions of the image unless they have a high signal energy (aka, they're significant in the reconstruction and representation o
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  • xc(t)=continuous time signal x[n]=discrete time signal
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  • *[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]] (used in [[ECE301]], [[ECE438]]) **[[CT Fourier Transform (frequency in radians per time unit)|Continuous-time Fourier Transform Pairs and Properties]] (function of radial frequency- in
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  • keywords: energy, power, signal '''Signal Metrics Definitions and Formulas'''
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  • Table of Continuous-time (CT) Fourier Transform Pairs and Properties | signal (function of t)
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  • ...ignals_ECE301S11|Compute the norm of a continuous-time complex exponential signal (practice problem)]] from [[ECE301]] ..._signals_ECE301S11|Compute the norm of a discrete-time complex exponential signal (practice problem)]] from [[ECE301]]
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  • =About the Continuous-time Fourier Transform= ...008mboutin| Example of how to take the Fourier transform of a non-periodic signal]]
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  • ...ler's formula to compute the norm of a continuous-time complex exponential signal (practice problem)]] from [[ECE301]] ...Euler's formula to compute the norm of a discrete-time complex exponential signal (practice problem)]] from [[ECE301]]
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  • ...ectrical Engineers, the first person that comes to mind when someone says "SIGNAL PROCESSING" is Fourier. *Jean Baptiste Joseph Fourier (1768 - 1830) laid a rock-solid foundation for signal analysis, when he claimed that all (continuously differentiable) signals ca
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  • We'll start with the 2-dimensional rect(*) signal. Here's the 2-dimensional rect(x,y) function and its CSFT:<br><br>
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  • ...your concepts of Fourier and Z transforms should be absolutely clear for signal processing (DSP ECE 438).--[[User:Hlalwani|Hersh Lalwani]] 14:55, 11 Decemb ...sforms such as laplace, furiere and z-transform and signals and systems of continuous-time and discrete-time. However, it contains a lot of mathematics skill and some
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  • As everybody in the class seems to know quite well, the continuous-time signal <math> x(t) = cos(t) </math> is a periodic function with period <math> 2\pi ...pling <math> x(t) </math> every <math>T</math>, we obtain a discrete-time signal <math>f[n]</math>. However, <math>f[n]</math> is not necessarily periodic:
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  • ...teral z-Transform. Sampling, quantization, and discrete-time processing of continuous-time signals. Discrete-time nonlinear systems: median-type filters, threshold de <br/>iii. an ability to determine the response of linear systems to any input signal by convolution in the time domain.
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  • <br/><br/>9. Signal detection, signal estimation, cross-correlation functions<br/><br/>
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  • [[Category:digital signal processing]] ...hat can be used to describe almost anything in the world be it an electric signal or the stock market. Did you know that our brain picks up different frequen
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  • Note: PM refers to the official course book, Digital Signal Processing, 3rd edition, J.G. Proakis and D.G. Manolakis. Prentice Hall, 19 *Basic Signals and Signal Properties
    9 KB (1,331 words) - 07:15, 29 December 2010
  • ...computation|A collective page to practice computing Fourier series of a CT signal]] ...putation_DT|A collective page to practice computing Fourier series of a DT signal]]
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  • Continuous-time Fourier Transform Pairs and Properties | signal (function of t)
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  • [[Category:Fourier series continuous-time]] ...pages contains exercises to practice computing the Fourier series of a CT signal =
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  • ...pages contains exercises to practice computing the Fourier series of a DT signal = *Fourier series of a discrete-time signal x[n] periodic with period N
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  • =Exercise: Compute the Fourier series coefficients of the following signal:= [[Recommended_exercise_Fourier_series_computation|More exercises on computing continuous-time Fourier series]]
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  • ...condition, the sampling frequency must be larger than the twice of maximum signal frequency, in order to avoid the aliasing when sampling. The frequency domain relationship betweeen a signal <math>x[n]</math> and its upsampled version <math>z[n]</math> can be shown
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  • ...es, a low-pass filter could be applied to this upsampling so to obtain the signal
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  • [[Category:digital signal processing]] <math>\text{ Sample a continuous signal x(t)=sin(}\omega t)\text{ with period of T, }</math>
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  • The highest frequency of the continuous signal is <math>f=\frac{\omega}{2\pi}</math>
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  • =Exercise: Compute the Fourier series coefficients of the following periodic signal:= ...ary to state the period in the question, as one can figure it out from the signal itself. </span> --[[User:Mboutin|Mboutin]] 08:15, 29 September 2010 (UTC)
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  • *Fourier series of a continuous-time signal x(t) periodic with period T *Fourier series coefficients of a continuous-time signal x(t) periodic with period T
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  • *Fourier series of a continuous-time signal x(t) periodic with period T *Fourier series coefficients of a continuous-time signal x(t) periodic with period T
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  • ...transforms one complex-valued function of a real variable into another. In signal processing, the domain of the original function is typically in the time do
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  • Definition: let x[n] be a DT signal with Period N. Then, ...efficients of continuous periodic function x[n]? <br> The DFT of a sampled signal x[n] of length N is directly proportional to the Fourier series coefficient
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  • [[Category:signal]] keywords:signal energy, exercises
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  • [[Category:signal]] =Continuous-Time (Average) Signal Power=
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  • ...ar filtering view of the STFT. Then we obtained a formula to reconstruct a signal from its STFT. This concluded the material on speech processing. We then be
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  • =About the Multiplication Property of the continuous-time Fourier transform= The multiplication property of the continuous-time Fourier transform can be stated as follows:
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  • ...[Complex Exponential and Sinusoidal Amplitude Modulation|videos explaining signal modulation]]! ...problems to practice CT convlution, and two problems for practicing basic signal's properties. -pm
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  • Compute the Magnitude of the following continuous-time signals ...of the result. (This is basically what you are doing in a), but since the signal is real, it is equal to its conjugate.) A quick note though on the symbol <
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  • [[Category:signal]] [[Category:continuous-time signal]]
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  • ...\omega\right| > \omega_m </math>. Can one recover the signal x(t) from the signal <math class="inline"> y(t)=x(t) p(t-3) </math>, where ...omega)\right| > \omega_m </math>. Can one recover the signal x(t) from the signal <math class="inline"> y(t)=x(t) p(t-3) </math>, where
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  • *[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]] *[[CT Fourier Transform (frequency in radians per time unit)|Continuous-time Fourier Transform Pairs and Properties]] (function of <span class="texhtml"
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  • ...olving|Practice Question]] on Computing the Fourier Series continuous-time signal= Obtain the Fourier series the CT signal
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  • *[[CT_Fourier_series_practice_problems_list|Problems on continuous-time Fourier series]] *[[CT_Fourier_transform_practice_problems_list|Problems on continuous-time Fourier transform]]
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  • ...ractice Question]] on Computing the Fourier Transform of a Continuous-time Signal = Compute the Fourier transform of the signal
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  • Today we obtained the formula for the Fourier transform of a periodic signal. We found that we cannot compute the Fourier transform of such signals usi We finished the lecture by discussing a few properties of the continuous-time Fourier transform.
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  • ...ractice Question]] on Computing the Fourier Transform of a Continuous-time Signal = Compute the Fourier transform of the signal
    2 KB (276 words) - 10:25, 11 November 2011
  • ...ractice Question]] on Computing the Fourier Transform of a Continuous-time Signal = Compute the Fourier transform of the signal
    2 KB (355 words) - 10:26, 11 November 2011
  • ...ractice Question]] on Computing the Fourier Transform of a Continuous-time Signal = Compute the Fourier transform of the signal
    853 B (122 words) - 10:26, 11 November 2011
  • ...ractice Question]] on Computing the Fourier Transform of a Continuous-time Signal = Compute the Fourier transform of the signal
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  • Compute the Fourier transform of the continuous-time signal <math>x(t)=e^{-3 |t|}</math>. (Use the definition of the Fourier transform, Compute the Fourier transform of the signal
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  • ...irst part of today's lecture, we finished discussing the properties of the continuous-time Fourier transform. We then used these properties to obtain a simple express ...of shifted copies (shifted by <math>2 \pi k</math>, with k integer)of our signal.
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  • = [[:Category:Problem_solving|Practice Question]] on the Properties of the Continuous-time Fourier Transform = Let x(t) be a continuous time signal with Fourier transform <math class="inline">{\mathcal X} (\omega) </math>.
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  • .../span> for the computation of the Fourier transform of the continuous-time signal. Check every step of the computation and remove point for any mistakes. Tak ...r:red"> 15 pts </span> for the computation of the Fourier transform of the signal. Check every step of the computation and remove point for any mistakes. Tak
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  • ...(t). &nbsp;The physiological system being modeled, h(t), will convert this signal into output, y(t). &nbsp;For example, the blood clotting mechanism can be i ...rresponding to adenine, guanine, cytosine, and thymine.) Furthermore, this signal must be broken into distinct genes and decoded. &nbsp;These genes can be ma
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  • The signal <math class="inline"> x(t)= e^{j \pi t }\frac{\sin (\pi t)}{t} </math> is s ...omega \right| > \omega_m </math>. Can one recover the signal x(t) from the signal <math class="inline"> y(t)=x(t) p(t-3) </math>, where
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  • [[Category:signal processing]] = [[ECE438|ECE 438]]: Digital Signal Processing with Applications =
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  • [[Category:signal processing]] Note: PM refers to the official course book, Digital Signal Processing, 3rd edition, J.G. Proakis and D.G. Manolakis. Prentice Hall, 19
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' Topic: Continuous-time Fourier transform computation (in terms of frequency f in hertz)
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' Topic: Continuous-time Fourier transform: from omega to f
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' Topic: Continuous-time Fourier transform of a complex exponential
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  • ...ture 13, we discussed the possibility of an extra credit project involving signal resampling and filtering. ...p to figure out how to transform this signal into the (higher resolution) signal
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  • [[Category:signal processing]] Pick 5 different continuous-time signals x(t) (at least three of which should be band-limited, and at least
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  • ...computation|A collective page to practice computing Fourier series of a CT signal]] ...putation_DT|A collective page to practice computing Fourier series of a DT signal]]
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' Give examples of continuous-time signals that are band-limited. (Justify your claim that they are band-limit
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  • '''Signal''' ...is a function, so when we say a continuous time signal or a discrete time signal we really mean continuous time functions and discrete time functions.
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  • [[Category:signal processing]]
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' ...n the exam, you could use the second approach if you were given a table of continuous-time fourier transform in which the Fourier transform of an exponential is given
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]'''
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]'''
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  • [[Category:signal processing]]
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  • ...tal_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]''' Compute the discrete-space Fourier transform of the following signal:
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  • [[Category:Digital Signal Processing]] ...://www.projectrhea.org/learning/practice.php Practice Problems] on Digital Signal Processing
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • = [[ECE]] 538: Digital Signal Processing I = *[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]]
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  • Communication, Networking, Signal and Image Processing (CS) ...that <math class="inline">\mathbf{X}\left(t\right)</math> is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function <math class="in
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS) Let <math class="inline">\mathbf{X}\left(t\right)</math> be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely
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  • Communication, Networking, Signal and Image Processing (CS) ...0"><span style="font-size: 19px;"><math>\color{blue} \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it
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  • Communication, Networking, Signal and Image Processing (CS) \text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by}
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  • Communication, Networking, Signal and Image Processing (CS) ...000"><span style="font-size: 19px;"><math>\color{blue}\text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it
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  • Communication, Networking, Signal and Image Processing (CS) \text{Let } F(\mu,\nu) \text{ be the continuous-time Fourier transform of } f(x,y) \text{ given by}
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  • *[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]] *[[CT Fourier Transform (frequency in radians per time unit)|Continuous-time Fourier Transform Pairs and Properties]] (function of <span class="texhtml"
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  • To measure the worst case time delay from accelerometer signal input to orientation angle output at a macro scale (ie not by counting asse ...S. Smith, "DSP Software," ''The Scientist and Engineer's Guide to Digital Signal Processing''. <http://www.dspguide.com/ch4.htm>
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  • **[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]] **[[CT_Fourier_Transform_(frequency_in_radians_per_time_unit)|Continuous-time Fourier Transform Pairs and Properties]] (function of <span class="texhtml"
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  • ...) = e^{-2}</math> at n=2, and so on. But because we have a constant input signal that effectively serves as a delta function at each time step, another impu ''<center>time-shifted continuous-time impulse responses due to a step function. (theoretically there should be an
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  • ...domain. It’s the output of In a LTI system when presented with a impulse signal input δ(t). In a LTI systems, impulse response is also equivalent to green
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  • ...hat the mean of the output random signal is equal to the mean of the input signal multiplied by the frequency response of the system evaluated at f=0. We als Note that we are now focusing only on continuous-time random processes for lack of time.
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  • ::↳ Eigen-Signal Analysis and Examples ...T is for continuous time signals. Let <math>x(n)</math> be a discrete time signal. Then, its DTFT, <math>X(e^{j\omega})</math> is given by <br/>
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • **[[SignalMetricsFormula|Signal Metrics Definitions and Formulas]] **[[CT_Fourier_Transform_(frequency_in_radians_per_time_unit)|Continuous-time Fourier Transform Pairs and Properties]] (function of <span class="texhtml"
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  • [[Category:signal processing]] ...in terms of <math>\omega</math>, which you have seen in [[ECE301]], to the continuous-time Fourier transform in terms of f. We then saw a few important properties of
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  • *[[Audio_Signal_Filtering|Audio Signal Filtering]]
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  • on continuous-time Fourier transform ==Collectively solved problems on continuous-time Fourier transform==
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  • on continuous-time Fourier series ==Collectively solved problems on continuous-time Fourier series==
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  • ...x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect t ...imited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula;
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  • [[Category:signal processing]] ''' [[ECE438| ECE438: Digital Signal Processing with Applications]]'''
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  • [[Category:signal processing]] ''' [[ECE438| ECE438: Digital Signal Processing with Applications]]'''
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  • [[Category:signal processing]] *[[2D_rect|2D rect signal]]
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  • [[Category:signal processing]] *[[2D_rect|2D rect signal]]
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  • Communication, Networking, Signal and Image Processing (CS)
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  • <font size="4">Communication, Networking, Signal and Image Processing (CS)</font>
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  • ** Be able to calculate the Fourier series coefficients of a period CT signal (DT Fourier series will NOT be on the exam). (3.28a(subparts abc), 3.22, 3. ...the frequency response of a system, find the FS coefficients of the output signal. (3.13, 3.14, Quiz 3)
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  • =Lab Wiki: [[2014_Fall_ECE_438_Boutin|ECE 438: Digital Signal Processing With Applications]], Fall 2014=
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • [[Category:signal processing]] ...in terms of <math>\omega</math>, which you have seen in [[ECE301]], to the continuous-time Fourier transform in terms of f. We then saw a few important properties of
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  • [[Category:signal processing]] ...led'. But this process is a little different from sampling. The continuous signal will become discrete after sampling while it will still be continuous after
    5 KB (768 words) - 18:51, 16 March 2015
  • ...nderstand the relation between comb vs original signal and rep vs original signal.
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  • ...x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect t ...imited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula;
    3 KB (462 words) - 09:43, 18 September 2014
  • ...x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect t ...imited interpolation of the samples (i.e., an expression for a continuous signal z(t) in terms of the samples y[n]). (Do not simply write down the formula;
    7 KB (1,178 words) - 20:16, 18 December 2014
  • [[Category:signal processing]] The discrete time fourier transform (DTFT) of a finite energy aperiodic signal x[n] can be given by the equation listed below. IT is a representation in t
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  • [[Category:signal processing]] ...g this cosine with period T (frequency <math>f_{s}=1/T</math>, the sampled signal <math>x[n]</math> can be written as <math>x[n]=cos(\frac{2 \pi f n}{T})</ma
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  • [[Category:signal processing]] ...nly be reconstructed perfectly from its sampling <math>x_s(t)</math>if the continuous-time Fourier Transform (CTFT) of <math>x(t)</math> is zero for all magnitudes of
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  • [[Category:signal processing]] ...cy domain view of the relationship between a signal and a sampling of that signal
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  • [[Category:signal processing]] ...cy domain view of the relationship between a signal and a sampling of that signal
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  • ...cy domain view of the relationship between a signal and a sampling of that signal </font> ...ship between a signal, and a sampling of that signal. Essentially, given a signal x(t), we are going to take a look at the similarities and differences in X(
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  • #Derivation of DTFT&nbsp;of downsampled signal<br> ...ure provides definition of downsampling, derives DTFT of&nbsp; downsampled signal and demonstrates it in a frequency domain. Also, it explains process of dec
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  • ...to eliminate the possibility of aliasing and distorting the reconstructed signal. Begin with x(t) as a continuous time signal with <math>x_1[n]= x(T_1*n)</math> being its discrete time sampling.
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  • ...cy domain view of the relationship between a signal and a sampling of that signal </font> ...f original signal x(t), sampled signal&nbsp; x<sub>s</sub>(t) and discrete signal x<sub>d</sub>[n]. Graphs in frequency domain&nbsp; help to understand this
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  • =Lab Wiki: [[2015_Fall_ECE_438_Boutin|ECE 438: Digital Signal Processing With Applications]], Fall 2015=
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • =Lab Wiki: [[2015_Spring_ECE_438_Ersoy|ECE 438: Digital Signal Processing With Applications]], Spring 2015=
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS) Let <math class="inline">\mathbf{X}\left(t\right)</math> be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely
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  • Communication, Networking, Signal and Image Processing (CS) ...that <math class="inline">\mathbf{X}\left(t\right)</math> is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function <math class="in
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS) ...that <math class="inline">\mathbf{X}\left(t\right)</math> is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function <math class="in
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  • Communication, Networking, Signal and Image Processing (CS)
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  • Communication, Networking, Signal and Image Processing (CS) Assume that <math>\mathbf{X}(t)</math> is a zero-mean continuous-time Gaussian white noise process with autocorrelation function
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  • [[Category:signal]] [[Category:continuous-time signal]]
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  • Chap 4. Continuous-Time Fourier Transform : [[Media:CTFT_Table.pdf|Continuous-Time Fourier Transform Table]] / [[Media:freq_allocation.pdf|US Frequency Alloca
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  • [[Category:signal processing]] ...tal signals and digital systems. We looked at some applications of digital signal processing, including
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  • ...e second goal of this homework is to learn different ways to reconstruct a signal. Consider the signal <math>x(t)=5 \text{sinc } ( \frac{t-7}{2} ).</math>
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  • ...be accomplished in an equivalent fashion by processing a sampling of that signal. ...t off frequency of 800Hz and a gain of 7. Let's call this desired filtered signal y(t).
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  • ...t off frequency of 800Hz and a gain of 7. Let's call this desired filtered signal y(t). ...such a way that a band-limited interpolation of the processed (output) DT signal would be the same as y(t)? Answer yes/no. If you answered yes, explain how.
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  • a) Obtain the Fourier transform X(f) of the signal and sketch the graph of |X(f)|. b) What is the Nyquist rate <math>f_0</math> for this signal?
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  • ...''aliasing'' that occurs when converting a continuous signal to a discrete signal via sampling. ...rmation applied to an aperiodic, discrete signal in order to represent the signal in terms of it's various spectral (frequency) components. It is defined as:
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • =Lab Wiki: [[2016_Fall_ECE_438_Boutin|ECE 438: Digital Signal Processing With Applications]], Fall 2016=
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  • The goal of this homework is to learn two different ways to reconstruct a signal. ...x(t) be a continuous-time signal and let y[n]=x(nT) be a sampling of that signal with period T>0. We would like to interpolate the samples (i.e., "connect t
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  • *go over the relationship between DT signal processing and CT signal processing (for a simple filter) once more, this time with a focus on the e Let x[n] be a DT signal. Let z[n]=x[2n] be a downsampling of x[n]. Let y[n] be an upsampling of x[n
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • =Lab Wiki: [[2017_Spring_ECE_438_Boutin|ECE 438: Digital Signal Processing With Applications]], Spring 2017=
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  • :b) understand the two different signal reconstruction methods we saw in class. ...ignal. Assuming that the highest frequency in the electroelectrocardiogram signal are at 2200 Hz, what criteria would you use to select the sampling frequen
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  • 3. A continuous time signal is such that <math> \chi(f) </math> is 0 when |f| > 3 KHz. You would like a ...mple exists with a sampling rate of 9000 samples/sec. Can you process this signal to make a band limited interpolation? If no, explain why. If yes, explain h
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  • ...region of convergence and translating a discrete signal into a continuous signal relate directly to the Laurent Series. ...like the Taylor Series, it builds a continuous function out of a discrete signal. This is the main way that the Laurent Series is related to DSP: its abilit
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  • =Lab Wiki: [[2017_Fall_ECE_438_Boutin|ECE 438: Digital Signal Processing With Applications]], Fall 2017=
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  • ...ncy-domain, one is primarily concerned with the information contained in a signal's component sinusoids. ...e integrated impulse response and will show how the filter will affect the signal in the time-domain. On the other hand, the frequency response will show how
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  • [[Category:signal]] [[Category:continuous-time signal]]
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  • [[Category:signal]] [[Category:continuous-time signal]]
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  • Biomedical signal processing aims at extracting significant information from physiological si ...sease, and earlier detection of both heart attacks and strokes. Biomedical signal processing is most useful in the critical care setting due to patient data
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  • ...quires Nlog(N). DFT is not the same as DTFT. Both start with discrete-time signal, but DFT produces a discrete frequency domain representation while the DTFT ...analysis. A common use of FFT’s is to find the frequency component of a signal buried in a noisy time domain. Given the MATLAB code bellow, a graph will b
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  • Communications and Signal Processing
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  • ...ideas developed in Calculus courses and expands on them with the focus on signal observation, analysis, and creation. It is unlike any other course I have t
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  • ...ideas developed in Calculus courses and expands on them with the focus on signal observation, analysis, and creation. It is unlike any other course I have t
    4 KB (644 words) - 11:22, 30 April 2019
  • =Lab wiki for ECE 438:Digital Signal Processing With Applications, Fall 2019=
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  • [[Category:digital signal processing]] [[Category:signal processing]]
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  • ...ignal directly, this is not possible in reality since storing a real-world signal would require an infinite amount of memory. Consequently, all signals are s With this information, we can express the sampling of a CT signal <math>x(t)</math> in terms of its sampling period T:
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  • The convolution forms the backbone of signal processing, but what are some direct applications of it? In this page, we w
    7 KB (1,006 words) - 22:10, 22 December 2019
  • ...lled amplitudes. In signal processing, there are discrete-time signals and continuous-time signals. When time is viewed as a discrete variable, then any function of t ...ractical cases, one can use Nyquist's Theorem to ensure that all essential signal information is collected, while not wasting effort measuring with too much
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