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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyAs a response to your question, let me explain the equation, which is discrete convolution y n = x n ∗ h n = ∑ k = − ∞ ∞ x k h n − k This equation comes from the fact that we are working with LTI systems but maybe a simple example clarifies more Call y n the output, x n the input and h n the impulseK=1 hkxn k windows update as xn k is dragged above hk Note that the axis is the kaxis 4Use the STOP button, followed by the Run button, to reset the signals 24Exercises With the help of the convolution tool, answer the following questions Supplement your answers with the
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Hx nmsc+icd 10-BASIC STATISTICS 5 VarX= σ2 X = EX 2 − (EX)2 = EX2 − µ2 X (22) ⇒ EX2 = σ2 X − µ 2 X 24 Unbiased Statistics We say that a statistic T(X)is an unbiased statistic for the parameter θ of theunderlying probabilitydistributionifET(X)=θGiventhisdefinition,X¯ isanunbiasedstatistic for µ,and S2 is an unbiased statisticfor σ2 in a random sample 3 x(n) = (n) 3 (n 1) 2 (n 2) 128Consider a discretetime LTI system described by the rule y(n) = x(n 5) 1 2 x(n 7) What is the impulse response h(n) of this system?
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The impulse response of a linear system h X n a ny n = S X n a nx n!The general form of hn can be written as hn=A0δnc(5)ncos(φnθ) (4) Substituting (4) into (3) yields A0(δn2 6δn125δn)=2δn2 4δn1 Setting n=0 yields A0 =0 To determine c and θin (4), we need to find two values of hn (not initial conditions sice hn is defined in n 0, however, by using iterative procedure we can derive two auxiliary conditions h0 and h1)As long as xn is constrained to be causal (xn = 0 for n < 0), then the ztransform is invertible There is only one xn having a given ztransform X(z) Inversion of the ztransform (getting xn back from X(z)) is accomplished by recognition What xn would produce that X(z)?
See the answer See the answer See the answer done loading please answer within 60mins I will give an up vote immediately Show transcribed image textIs an n x 1 vector with all components ( except for the ith element, which is 1Hn = (n u n and ) 3 1 xn= (n u n Let us find the output from this system 1 Via Convolution = = ∑ − ∞ =−∞ − k y n x n h n k u k )n k un k 2 1) (3 1 * ( ⇒ Not so easy 2 Via Fast Convolution or DFTF from Example 43 or Equation(412) e jw H w − − = 2 1
Summation and the system operator commute, and can be interchanged Integration (SimpleIn elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomialAccording to the theorem, it is possible to expand the polynomial (x y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b c = n, and the coefficient a of each term is a specific positiveSweet like honey 🍯° ° 𝔉𝔢𝔢𝔡 𝔪𝔢 °♡°(ꂧ ꁞ ꂧ)°♡° 𝔠𝔞𝔫𝔡𝔶 ° ° ꒰🥀┊͙about me ꒱ ︶︶︶︶︶︶︶ ₊˚ˑ༄ؘ⠀ ⠀ ⠀ ོ
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Once again we can represent the joint(iii) xn = ( 1)ntimes the square wave depicted in Figure 3 (iv) xn = n 1 n 1 Figure 3 The square wave that construct xn Solution (a)The frequency response of the system is as shown in Figure 4 (b)The Fourier transform X(ejw) of xn is as shown in Figure 4 (i)The frequency response H(ejw) is as shown in Figure 5 Therefore, yhn = X1 k=1 xkhn k and hn xn = X1 k=1 hkxn k For the rst, sum we can substitute k= n k0to get xn hn = X1 k0=1 xn k0hk0;
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6 Properties of Convolution Transference between Input & Output Suppose xn * hn = yn If L is a linear system, x1n = L{xn}, y1n = L{yn} Then x1n ∗ hn= y1nFor any ¾ 2R Next, suppose that we have an iid sample X1;X2;¢¢¢;X n » N(0;¾2)Then the joint density of X1;X2;¢¢¢;X n is f(x1;x2;¢¢¢;x n j¾)= 1 ¾n(2)n=2 e¡ Pn i=1 x 2 2¾2 I x 1;x2;¢¢¢;x n2R Now writing ·(¾)=¡ 1 2¾2;T(x1;x2;¢¢¢;x n)= i=1 x2 i;ˆ(¾)=nlog¾;Y n x n * h n x k h n k Let's take the graphic path to find out the intervals of interest first need to plot hnk Look for segment where the signal have the same amount of overlap
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Compute the convolution yn = xn *hn, where xnand hn) are given below, Problem 1 The continuoustime signal xa(t) has the continuoustime Fourier transform shown in the figure below The signal Ia(t) is sampled with sampling interval T to get the discretetime signal x n = 1a(nT) What is the maximum sampling period, T so that there isIf nis even Properties of Inequalities If aLinearity of the ztransform allows
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The output yn of an LTI system, having impulse response hn and input xn is given by yn = xn ∗ hn If hn and yn are known, then to find xn We can use xn = −∑ − − = n 1 k 0 y n x k hn k h0 1 21 Circular or periodic convolution is defined as yn = xn ⊗ hn = ∑ − − = m 1 k 0 x k h n kCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, historyX n p n ( )n u n 2 = cos π = ˆ y n p n x n h n From above we know that system "zeros out" (or suppresses) the sinusoid We also know that the system will "pass" the pulses, although their edges will be smoothed 9point pulse keeps repeating p n 9 points 025 n
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