Chapter 6: Applications of Fourier Representation Houshou Chen
|
|
- Branden Byrd
- 5 years ago
- Views:
Transcription
1 Chapter 6: Applications of Fourier Representation Houshou Chen Dept. of Electrical Engineering, National Chung Hsing University
2 H.S. Chen Chapter6: Applications of Fourier Representation 1 Applications of FS, DTFS, FT, and DTFT In the previous chapters, we developed the Fourier representations of four distinct classes of signals. 1. FS for periodic continuous-time signals. 2. DTFS for periodic discrete-time signals. 3. FT for aperiodic continuous-time signals. 4. DTFT for aperiodic discrete-time signals. In this chapter, we consider mixed signals such as 1. periodic and aperiodic signals 2. continuous- and discrete-time signals
3 H.S. Chen Chapter6: Applications of Fourier Representation 2 If we apply a periodic signals to a stable LTI system, the convolution operation involves a mixing of aperiodic impulse response and periodic input. A system that samples continuous-time signals involves both continuous- and discrete-time signals. In order to use Fourier methods to analyze such interactions, we must build bridges between the Fourier representation of different classes of signals.
4 H.S. Chen Chapter6: Applications of Fourier Representation 3 We can develop FT and DTFT representations of continuous- and discrete-time periodic signals, respectively. FT can also be used to analyze problems involving mixtures of continuous- and discrete-time signals. FT and DFTT are most commonly used for analysis applications. The DTFS is the primary representation used for computational applications. We will consider the sampling theorem and FFT in this chapter. A thorough understanding of the relationship between the four Fourier representations is a critical step in using Fourier methods to solve problems involving signals and systems.
5 H.S. Chen Chapter6: Applications of Fourier Representation 4 Strictly speaking, neither the FT nor the DTFT converges for periodic signals. However, by incorporating impulse into the FT and DTFT, we may develop FT and DTFT representation for periodic signals. We may use them and the properties of FT and DTFT to analyze problems involving mixtures of periodic and aperiodic signals. We will consider the convolutional and multiplication of aperiodic and periodic signals in time domain and see what happens in frequency domain.
6 H.S. Chen Chapter6: Applications of Fourier Representation 5 Relating FT and FS Given a continuous-time periodic signal x(t) with FS representation x(t) = k= X[k]e jkw 0t. Recall the following FT pair (with impulse in frequency domain) e jkw 0t FT 2πδ(w kw 0 ) We thus have the FT for x(t) as follows. x(t) = k= X[k]e jkw 0t FT X(jw) = 2π k= X[k]δ(w kw 0 )
7 H.S. Chen Chapter6: Applications of Fourier Representation 6 Thus, the FT of a periodic signal is a series of impulses spaced by the fundamental frequency w 0. The kth impulse has strength 2πX[k], where X[k] is the kth FS coefficient.
8 H.S. Chen Chapter6: Applications of Fourier Representation 7
9 H.S. Chen Chapter6: Applications of Fourier Representation 8 Find the FT representation of x(t) = cos w 0 t
10 H.S. Chen Chapter6: Applications of Fourier Representation 9 The FT of a unit impulse train p(t) = n= δ(t nt) Since p(t) is periodic with fundamental frequency w 0 = 2π/T and the FS coefficients are given by p[k] = 1/T T/2 T/2 δ(t)e jkw 0t dt = 1/T. Therefore the FT of p(t) is given by P(jw) P(jw) = 2π T k= δ(w kw 0 ). Hence, the FT of p(t) is also an impulse train; that is, an impulse train is its own FT.
11 H.S. Chen Chapter6: Applications of Fourier Representation 10 The spacing between the impulses in the frequency domain is inversely related to the spacing between the impulses in the time domain.
12 H.S. Chen Chapter6: Applications of Fourier Representation 11
13 H.S. Chen Chapter6: Applications of Fourier Representation 12 Relating DTFT and DTFS Given a discrete-time periodic signal x[n] with DTFS representation x[n] = N 1 k=0 X[k]e jkw 0n. As in the FS case, the key observation is that the inverse DTFT of a frequency-shifted impulse is a discrete-time complex sinusoid. The DTFT is a 2π-periodic function of frequency, so we may express for w [ π, π] and kw 0 [ π, π] e jkw 0n DTFT δ(w kw 0 )
14 H.S. Chen Chapter6: Applications of Fourier Representation 13 Or as the following DTFT pair (with impulse in frequency domain) e jkw 0n DTFT m= 2πδ(w kw 0 m2π)
15 H.S. Chen Chapter6: Applications of Fourier Representation 14 We thus have the DTFT for x[n] as follows. x[n] = N 1 k=0 X[k]e jkw 0n N 1 DTFT X(e jw ) = 2π Or equivalently as follows. k=0 X[k] m= δ(w kw 0 m2π) x[n] = N 1 k=0 X[k]e jkw 0n DTFT X(e jw ) = 2π m= X[k]δ(w kw 0 )
16 H.S. Chen Chapter6: Applications of Fourier Representation 15 Thus, the DTFT of a periodic signal is a series of impulses spaced by the fundamental frequency w 0. The kth impulse has strength 2πX[k], where X[k] is the kth DTFS coefficient.
17 H.S. Chen Chapter6: Applications of Fourier Representation 16
18 H.S. Chen Chapter6: Applications of Fourier Representation 17 Convolution of periodic and aperiodic signals Use the fact that convolution in the time domain corresponds to multiplication in the frequency domain. That is, y(t) = x(t) h(t) FT Y (jw) = X(jw)H(jw). If the input x(t) is periodic with period T, then x(t) FT X(jw) = 2π k= where X[k] are the FS coefficients of x(t). X[k]δ(w kw 0 ), We substitute this representation into the convolution property to obtain y(t) = x(t) h(t) FT Y (jw) = 2π k= H(jkw 0 )X[k]δ(w kw 0 )
19 H.S. Chen Chapter6: Applications of Fourier Representation 18 The form of Y (jw) implies that y(t) corresponds to a periodic signal with the same period T as x(t) Indeed, The strength of the kth impulse in X(jw) is adjusted by the value of H(jw) evaluated at the frequency at which it is located, or H(jkw 0 ), to yield an impulse in Y (jw) at w = kw 0. The results show that the periodic extension in time domain corresponds to the discrete operation in frequency domain.
20 H.S. Chen Chapter6: Applications of Fourier Representation 19
21 H.S. Chen Chapter6: Applications of Fourier Representation 20 Example: Let the input signal applied to an LTI system with impulse response h(t) = 1/(πt) sin(πt) be the periodic square wave. Find the output of the system. The frequency response of h(t) can be shown to be low pass filter h(t) FT 1, w π H(jw) = 0, w > π The FT of square wave can be found by the FS coefficients as follows. x(t) FT X(jw) = k= 2 sin(kπ/2) δ(w kπ/2) k
22 H.S. Chen Chapter6: Applications of Fourier Representation 21 There are five terms of X(jw) inside [ π, π], i.e., k = 2, 1, 0, 1,, 2, 2 sin( 2π/2) sin( 0π/0) 0 δ(w ( 2)π/2) + 2 sin( 1π/2) δ(w ( 1)π/2) ( 1) δ(w 0π/2)+ 2 sin(1π/2) 1 = 2δ(w + π/2) + πδ(w) + 2δ(w π/2) δ(w 1π/2)+ 2 sin(2π/2) δ(w 2π/2) 2 Finally, Y (jw) is obtained by the fact that H(jw) acts as a low-pass filter, passing the harmonics at π/2, 0, and π/2, while suppressing all others. Y (jw) = πδ(w) + 2δ(w π/2) + 2δ(w + pi/2) Taking the inverse FT of Y (jw) gives the output. Thus y(t) = 1/2 + 2/π cos(π/2t)
23 H.S. Chen Chapter6: Applications of Fourier Representation 22
24 H.S. Chen Chapter6: Applications of Fourier Representation 23 Multiplication of periodic and aperiodic signals Recall the multiplication property of the FT, represented as y(t) = g(t)x(t) FT Y (jw) = 1 G(jw) X(jw). 2π If the signal x(t) is periodic with period T, then x(t) FT X(jw) = 2π Therefore, the FT of y(t) is k= y(t) = g(t)x(t) FT Y (jw) = G(jw) X[k]δ(w kw 0 ). k= X[k]δ(w kw 0 ). Finally, by the sifting property of the impulse function, the convolution of any function with a shifted impulse results in a shifted version of the
25 H.S. Chen Chapter6: Applications of Fourier Representation 24 original funciton, i.e., y(t) = g(t)x(t) FT Y (jw) = k= X[k]G(j(w kw 0 )). Multiplication of g(t) with the periodic function x(t) gives an FT consisting of a weighted sum of shifted version of G(jw). As expected, the form of Y (jw) corresponds to the FT of a continuous-time aperiodic signal, since the product of periodic and aperiodic signals is aperiodic.
26 H.S. Chen Chapter6: Applications of Fourier Representation 25
27 H.S. Chen Chapter6: Applications of Fourier Representation 26 Let x(t) be the continuous-time impulse train. x(t) = n= δ(t nt) Remark: By introducing the delta function in time domain, we can represent a discrete-time function as a continuous-time function. For example, the discrete-time periodic signal x[n] = 1, for all n corresponds to continuous-time periodic x(t) as above. The FT of x(t) is also a periodic impulse train in frequency domain X(jw) = 2π T k= δ(w kw 0 ).
28 H.S. Chen Chapter6: Applications of Fourier Representation 27 Now y(t) = g(t)x(t) is the sampled version of g(t) and the FT of y(t) is y(t) = g(t)x(t) FT Y (jw) = k= 2π T G(j(w kw 0)). We see that Y (jw) is the periodic extension of G(jw). The corresponded result is called the sampling theorem and we will discuss now. The results show that the discrete operation in time domain corresponds to the periodic extension in frequency domain
29 H.S. Chen Chapter6: Applications of Fourier Representation 28 Relating the FT to the DTFT We can derive an FT representation of discrete-time signals by incorporating impulses into the description of the signals in the time domain. Therefore, the FT is a powerful tool for analyzing problems involving mixtures of discrete- and continuous-time signals. Combine the results of the relationship between FT and FS, also FT and DTFT, the FT can be used for the four classes of signals.
30 H.S. Chen Chapter6: Applications of Fourier Representation 29 Consider the DTFT of an arbitrary discrete-time signals x[n]: X(e jω ) = n= x[n]e jωn. We seek an FT pair x δ (t) FT X δ (jw) that corresponds to the DTFT pair x[n] DTFT X(e jω ).
31 H.S. Chen Chapter6: Applications of Fourier Representation 30 Now, let Ω = wt s, where x[n] = x δ (nt s ). I.e., x[n] is equal to the samples of x(t) taken at intervals of T s. By this substitution, Ω = wt s, we transform X(e jω ) of Ω into X δ (jw) of w X δ (jw) = X(e jω ) Ω=wTs = x[n]e jwtsn. n= Taking the inverse FT of X δ (jw) and use the following fact δ(t nt s ) DTFT e jwt sn, we obtain the continuous-time x δ (t) x δ (t) = n= x[n]δ(t nt s ).
32 H.S. Chen Chapter6: Applications of Fourier Representation 31 Hence, we have x[n] DTFT X(e jω ) = n= x[n]e jωn and x δ (t) FT X δ (jw) = n= x[n]e jwt sn
33 H.S. Chen Chapter6: Applications of Fourier Representation 32
34 H.S. Chen Chapter6: Applications of Fourier Representation 33 Sampling Theorem : We use the FT representation of discrete-time signals to analyze the effects of uniformly sampling a signal. The sampling operation generates a discrete-time signal from a continuous-time signal. We will see the relationship between the DTFT of the sampled signals and the FT of the continuous-time signal. Sampling of continuous-time signals is often performed in order to manipulate the signal on a computer or microprocessor.
35 H.S. Chen Chapter6: Applications of Fourier Representation 34 A signal x(t) with X(jw) as follows X(jw ) - w b w b w is called band-limited signal and can be exactly reconstructed from its samples {x(nt S )} n= provided that the sampling frequency ω s = 2π T S 2ω B, 2ω B : Nyquist sampling rate.
36 H.S. Chen Chapter6: Applications of Fourier Representation 35 Ideal Sampling : X(t) Xs(t) H(jw)? Xr(t) = X(t) P(t) First, we multiply x(t) by the impulse train P(t) = n= δ(t nt s ) (period T s ) x s (t) = p(t)x(t) X(t) Xs(t) -2Ts -Ts 0 Ts 2Ts
37 H.S. Chen Chapter6: Applications of Fourier Representation 36
38 H.S. Chen Chapter6: Applications of Fourier Representation 37 Now, x s (t) = x(t)p(t) = x(t) δ(t nt s ) n= = x(t)δ(t nt s ) n = n= x(nt s )δ(t nt s ) Next, we want to find X s (jw) p(t) = δ(t nt s ) = C k e jkw st n= where C k = 1 T s <T s > = 1 T s Ts 0 n= P(t)e jkw st dt δ(t)e jkw st dt = 1 T s
39 H.S. Chen Chapter6: Applications of Fourier Representation 38 Since x s (t) = x(t)p(t) p(t) = P(jw) = n= n= 1 T s e jkw st 2π T s δ(w kw s ) X s (jw) = 1 X(jw) p(jw) 2π = 1 2π X(jw) = 1 T s k= = 1 T s k= k= 2π T s δ(w kw s ) X(jw) δ(w kw s ) X(j(w kw s ))
40 H.S. Chen Chapter6: Applications of Fourier Representation 39 The sampling theorem says that the discrete operation in time domain x s (t) = x(t)p(t) corresponds to the periodic extension in frequency domain X s (jw) = 1 T s X(j(w kw s )) k= X(jw) Xs(jw) 1 1/Ts -W B W B W B Ws
41 H.S. Chen Chapter6: Applications of Fourier Representation 40
42 H.S. Chen Chapter6: Applications of Fourier Representation 41 We can recover X(jw) from X s (jw) if and only if w s w B w B w s 2w B To recover x(t), we multiply X s (jw) by H(jw) Ts -W B W B w X r (jw) = X s (jw) H(jw)
43 H.S. Chen Chapter6: Applications of Fourier Representation 42 Since H(jw) = T s rect( w 2w B ) h(t) = T s 1 2π 2w Bsinc( 2w B 2 t) = T s w B π sinc(w Bt) = 2w B sinc(w B t) w s x r (t) = x s (t) h(t) = [ = n= n= x(nt s )δ(t nt s )] [ 2w B w s sinc(w B t)] x(nt s ) 2w B w s sinc(w B (t nt S )) = x(t) iff w s 2w B
44 H.S. Chen Chapter6: Applications of Fourier Representation 43 FFT (Fast Fourier Transform) The role of DTFS as a computational tool is greatly enhanced by the availability of efficient algorithms for evaluating the forward and inverse DTFS. We call these algorithms fast Fourier transform (FFT) algorithm. FFT use the divide and conquer principle by dividing the DTFS into a series of lower order DTFS and using the symmetry and periodicity properties of the complex sinusoid e jk2πn/n. The total computations of FFT is substantially less than the original DTFS.
45 H.S. Chen Chapter6: Applications of Fourier Representation 44
46 H.S. Chen Chapter6: Applications of Fourier Representation 45 The computation of x[n] from X[k] or the computation of X[k] from x[n] requires N 2 complex multiplications and N N complex additions. Assume N is a power of 2, we can thus split x[n], 0 n N 1, into even and odd indexed signals, i.e., x[2n] and x[2n + 1], 0 n N/2 1.
47 H.S. Chen Chapter6: Applications of Fourier Representation 46 (1) Decimation in time F k = N 1 n=0 2πnk j f n e N DFT if N is a power of 2 e.q. {f n } = {f 0, f 1, f 2, f 3, f 4, f 5, f 6, f 7 } N = 8 Define g n = f 2n (even-number samples) h n = f 2n+1 (odd-number samples) n = 0, 1,, N 2 1
48 H.S. Chen Chapter6: Applications of Fourier Representation 47 N F k = = = { N 1 n=0 N 2 1 n=0 N 2 1 n=0 2πnk j f n e N = DFTN {f n } 2π(2n)k j f 2n e N + 2πnk j g n e N 2 1 n=0 N N/2 } + { 2 1 n=0 2π(2n+1)k j f 2n+1 e N 2πnk j h n e N/2 }e j 2πk N 2πk j = DFT N/2 {g n } + e N DFTN/2 {h n } Define W N = e j 2π N W k N 2πk = e j N NF k = [ N 2 NF k = [ N 2 G k N 2 G k ] + W k N[N 2 ] + W k N[N 2 H k ] 0 k N 2 1 H k N 2 ] N 2 k N 1
49 H.S. Chen Chapter6: Applications of Fourier Representation 48 In summary, we have the following = NF k = [ N 2 G k ] + W k N [ N 2 NF k = [ N 2 G k N 2 H k ] 0 k N 2 1 ] + W k N [ N 2 H k N 2 ] N 2 k N 1 This indicates that F[k] and F[k + N/2], 0 k N/2 1, are a weighted combination of G[k] and H[k] This structure is called a butterfly structure.
50 H.S. Chen Chapter6: Applications of Fourier Representation 49 For example : N=8 k = 1 8F 1 = 4 G 1 + W 1 8 ( 4 H 1 ) k = 5 8F 5 = 4 G 1 + W 5 8 ( 4 H 1 ) G 0 G 1 1 F 0 F 1 G 2 G 3 W 8 1 F 2 F 3 This structure is call Butterfly =2 complex adds +1 complex multiplication H 0 1 F 4 H 1 H 2 H 3 W 8 1 =W 8 4 W 8 1 = -W 8 1 F 5 F 6 F 7
51 H.S. Chen Chapter6: Applications of Fourier Representation 50 We can further simplify the results by exploiting the following fact. For k N/2, we have W k N = W N 2 N W k N 2 N = W k N 2 N W N 2 N = e j 2π N N 2 = e jπ = 1
52 H.S. Chen Chapter6: Applications of Fourier Representation 51 For example : 8-point FFT
53 H.S. Chen Chapter6: Applications of Fourier Representation 52
54 H.S. Chen Chapter6: Applications of Fourier Representation 53
55 H.S. Chen Chapter6: Applications of Fourier Representation 54
56 H.S. Chen Chapter6: Applications of Fourier Representation 55
57 H.S. Chen Chapter6: Applications of Fourier Representation 56
58 H.S. Chen Chapter6: Applications of Fourier Representation 57 Operation Counted in FFT algorithm/dft algorithm Total operations in FFT ( 2 M = N) =(M sections) (N/2 butterfly/section) (3 operations/butterfly) = 3 2 NM = 3 2 N log 2N
59 H.S. Chen Chapter6: Applications of Fourier Representation 58 Decimation in Frequency F k = N 1 n=0 2πnk j f n e N if N is a power of 2,eg. N=8 Def ine {f n } = {f 0, f 1, f 2, f 3, f 4, f 5, f 6, f 7 } g n = f n ( first half samples ) n = 0, 1,, N 2 1 h n = f n+ N 2 ( second half sample )
60 H.S. Chen Chapter6: Applications of Fourier Representation 59 F k = = = N 2 1 n=0 N 2 1 n=0 N 2 1 2πnk j f n e N 2πnk j g n e N N n=0 N n=0 (g n + h n e jπk ) e n=0 f n+ N 2 2πnk j h n e N 2πnk j N e j 2π(n+ N )k 2 N e jπk N=8 {F k } = {F 0, F 1, F 2, F 3, F 4, F 5, F 6, F 7 } Define 2 sequences {R k } = {F 0, F 2, F 4, F 6 }even samples {S k } = {F 1, F 3, F 5, F 7 }odd samples
61 H.S. Chen Chapter6: Applications of Fourier Representation 60 even number group k = 2k F 2k = = N 2 1 (g n + h n e } jπ 2k {{} n=0 =1 N 2 1 (g n + h n )e n=0 j 2πnk )e j 2πn(2k ) N N/2 N 2 point DFT NF 2k = DFT N 2 {g n + h n } k = 0, 1,, N 2 1
62 H.S. Chen Chapter6: Applications of Fourier Representation 61 odd number group k = 2k + 1 F 2k +1 = = N 2 1 (g n + h n e jπ(2k +1) }{{} n=0 = 1 N 2 1 [(g n h n )e n=0 NF 2k+1 = DFT N 2 2πn j N )e j 2πn(2k +1) N ]e j 2πnk N/2 2πn j {(g n h n )e N } = NF 2k = DFT N 2 NF 2k+1 = DFT N 2 = DFT N 2 {(g n h n )W n N } k = 0, 1,, N 2 1 [g n + h }{{ n ] k = 0, 1,, N } 2 1 g n [(g n h n )WN n ] k = 0, 1,, N }{{} 2 1 h n
63 H.S. Chen Chapter6: Applications of Fourier Representation 62 We already reduce N-point DFT to N 2 -point DFT. we can repeat this process until we get one-point DFT if N is a power of 2. N=2 M f 0 g 0 g 0 f 1 g 1 1 g 1 g n + h n =g n fl f 2 g 2 1 g 2 (g n - h n n )W N = h n fl f 3 g 3 g 3 f 4 f 5 h 0 h 1-1W 1 h 0 h 1 Butterfly = 2 complex adds n +1 complex multiplication N ) (W f 6 h 2 h 2 f 7 h 3 h 3
64 H.S. Chen Chapter6: Applications of Fourier Representation 63 Example:
65 H.S. Chen Chapter6: Applications of Fourier Representation 64 Total operations in FFT (2 M = N) =( M sections) ( N 2 butterfly/section) (3 operations/butterfly) = 3 2 NM = 3 2 N log 2N Operations counted in FFT algorithm: total operations in DFT N 2 ( X k = N 1 n=0 x 2πnk j N ne, k = 0, 1,, N 1) total operations in FFT N log 2 N Improvement Ratio= DFT FFT = N log 2 N
Fourier series for continuous and discrete time signals
8-9 Signals and Systems Fall 5 Fourier series for continuous and discrete time signals The road to Fourier : Two weeks ago you saw that if we give a complex exponential as an input to a system, the output
More informationFourier Series and Fourier Transforms
Fourier Series and Fourier Transforms Houshou Chen Dept. of Electrical Engineering, National Chung Hsing University E-mail: houshou@ee.nchu.edu.tw H.S. Chen Fourier Series and Fourier Transforms 1 Why
More informationReview: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationComplex symmetry Signals and Systems Fall 2015
18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The
More informationECE 301: Signals and Systems Homework Assignment #7
ECE 301: Signals and Systems Homework Assignment #7 Due on December 11, 2015 Professor: Aly El Gamal TA: Xianglun Mao 1 Aly El Gamal ECE 301: Signals and Systems Homework Assignment #7 Problem 1 Note:
More informationDigital Signal Processing. Midterm 1 Solution
EE 123 University of California, Berkeley Anant Sahai February 15, 27 Digital Signal Processing Instructions Midterm 1 Solution Total time allowed for the exam is 8 minutes Some useful formulas: Discrete
More informationContinuous-Time Fourier Transform
Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT Lowpass Filtering Blurring or Smoothing Original
More informationECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,
More informationFast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding
Fast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding Fourier Series & Transform Summary x[n] = X[k] = 1 N k= n= X[k]e jkω
More informationFast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding
Fast Fourier Transform Discrete-time windowing Discrete Fourier Transform Relationship to DTFT Relationship to DTFS Zero padding J. McNames Portland State University ECE 223 FFT Ver. 1.03 1 Fourier Series
More information4.1 Introduction. 2πδ ω (4.2) Applications of Fourier Representations to Mixed Signal Classes = (4.1)
4.1 Introduction Two cases of mixed signals to be studied in this chapter: 1. Periodic and nonperiodic signals 2. Continuous- and discrete-time signals Other descriptions: Refer to pp. 341-342, textbook.
More informationSignals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions
8-90 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 08 Midterm Solutions Name: Andrew ID: Problem Score Max 8 5 3 6 4 7 5 8 6 7 6 8 6 9 0 0 Total 00 Midterm Solutions. (8 points) Indicate whether
More informationECE 301 Fall 2011 Division 1 Homework 10 Solutions. { 1, for 0.5 t 0.5 x(t) = 0, for 0.5 < t 1
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Let x be a periodic continuous-time signal with period, such that {, for.5 t.5 x(t) =, for.5
More informationELEN 4810 Midterm Exam
ELEN 4810 Midterm Exam Wednesday, October 26, 2016, 10:10-11:25 AM. One sheet of handwritten notes is allowed. No electronics of any kind are allowed. Please record your answers in the exam booklet. Raise
More informationDigital Signal Processing. Midterm 2 Solutions
EE 123 University of California, Berkeley Anant Sahai arch 15, 2007 Digital Signal Processing Instructions idterm 2 Solutions Total time allowed for the exam is 80 minutes Please write your name and SID
More informationFinal Exam of ECE301, Section 3 (CRN ) 8 10am, Wednesday, December 13, 2017, Hiler Thtr.
Final Exam of ECE301, Section 3 (CRN 17101-003) 8 10am, Wednesday, December 13, 2017, Hiler Thtr. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, and
More informationLecture 19: Discrete Fourier Series
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 19: Discrete Fourier Series Dec 5, 2001 Prof: J. Bilmes TA: Mingzhou
More informationFall 2011, EE123 Digital Signal Processing
Lecture 6 Miki Lustig, UCB September 11, 2012 Miki Lustig, UCB DFT and Sampling the DTFT X (e jω ) = e j4ω sin2 (5ω/2) sin 2 (ω/2) 5 x[n] 25 X(e jω ) 4 20 3 15 2 1 0 10 5 1 0 5 10 15 n 0 0 2 4 6 ω 5 reconstructed
More informationEE 224 Signals and Systems I Review 1/10
EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS
More informationFourier Transform for Continuous Functions
Fourier Transform for Continuous Functions Central goal: representing a signal by a set of orthogonal bases that are corresponding to frequencies or spectrum. Fourier series allows to find the spectrum
More informationBridge between continuous time and discrete time signals
6 Sampling Bridge between continuous time and discrete time signals Sampling theorem complete representation of a continuous time signal by its samples Samplingandreconstruction implementcontinuous timesystems
More informationX. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More informationDiscrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz
Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.
More information! Circular Convolution. " Linear convolution with circular convolution. ! Discrete Fourier Transform. " Linear convolution through circular
Previously ESE 531: Digital Signal Processing Lec 22: April 18, 2017 Fast Fourier Transform (con t)! Circular Convolution " Linear convolution with circular convolution! Discrete Fourier Transform " Linear
More informationECE 301: Signals and Systems Homework Assignment #5
ECE 30: Signals and Systems Homework Assignment #5 Due on November, 205 Professor: Aly El Gamal TA: Xianglun Mao Aly El Gamal ECE 30: Signals and Systems Homework Assignment #5 Problem Problem Compute
More informationSignals and Systems Spring 2004 Lecture #9
Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationOverview of Sampling Topics
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
More informationDiscrete Fourier Transform
Discrete Fourier Transform Virtually all practical signals have finite length (e.g., sensor data, audio records, digital images, stock values, etc). Rather than considering such signals to be zero-padded
More informationFinal Exam of ECE301, Prof. Wang s section 8 10am Tuesday, May 6, 2014, EE 129.
Final Exam of ECE301, Prof. Wang s section 8 10am Tuesday, May 6, 2014, EE 129. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e-mail address, and signature
More information6.003: Signal Processing
6.003: Signal Processing Discrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete-Time Fourier Transform (DTFT) Relations to Discrete-Time Fourier Series (DTFS) October 16, 2018
More informationUniversity Question Paper Solution
Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=
More informationChapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter
Chapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter Objec@ves 1. Learn techniques for represen3ng discrete-)me periodic signals using orthogonal sets of periodic basis func3ons. 2.
More informationChap 4. Sampling of Continuous-Time Signals
Digital Signal Processing Chap 4. Sampling of Continuous-Time Signals Chang-Su Kim Digital Processing of Continuous-Time Signals Digital processing of a CT signal involves three basic steps 1. Conversion
More informationDISCRETE FOURIER TRANSFORM
DISCRETE FOURIER TRANSFORM 1. Introduction The sampled discrete-time fourier transform (DTFT) of a finite length, discrete-time signal is known as the discrete Fourier transform (DFT). The DFT contains
More informationInterchange of Filtering and Downsampling/Upsampling
Interchange of Filtering and Downsampling/Upsampling Downsampling and upsampling are linear systems, but not LTI systems. They cannot be implemented by difference equations, and so we cannot apply z-transform
More informationRepresenting a Signal
The Fourier Series Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity and timeinvariance of the system and represents the
More information8 The Discrete Fourier Transform (DFT)
8 The Discrete Fourier Transform (DFT) ² Discrete-Time Fourier Transform and Z-transform are de ned over in niteduration sequence. Both transforms are functions of continuous variables (ω and z). For nite-duration
More informationAspects of Continuous- and Discrete-Time Signals and Systems
Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the
More informationReview of Discrete-Time System
Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.
More informationFourier Analysis Overview (0B)
CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission
More informationFrequency-Domain C/S of LTI Systems
Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the
More information/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by
Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,
More informationso mathematically we can say that x d [n] is a discrete-time signal. The output of the DT system is also discrete, denoted by y d [n].
ELEC 36 LECURE NOES WEEK 9: Chapters 7&9 Chapter 7 (cont d) Discrete-ime Processing of Continuous-ime Signals It is often advantageous to convert a continuous-time signal into a discrete-time signal so
More informationJ. McNames Portland State University ECE 223 Sampling Ver
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
More information6.003 Signal Processing
6.003 Signal Processing Week 6, Lecture A: The Discrete Fourier Transform (DFT) Adam Hartz hz@mit.edu What is 6.003? What is a signal? Abstractly, a signal is a function that conveys information Signal
More informationContinuous Fourier transform of a Gaussian Function
Continuous Fourier transform of a Gaussian Function Gaussian function: e t2 /(2σ 2 ) The CFT of a Gaussian function is also a Gaussian function (i.e., time domain is Gaussian, then the frequency domain
More informationFinal Exam of ECE301, Prof. Wang s section 1 3pm Tuesday, December 11, 2012, Lily 1105.
Final Exam of ECE301, Prof. Wang s section 1 3pm Tuesday, December 11, 2012, Lily 1105. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e-mail address,
More information6.003 Signal Processing
6.003 Signal Processing Week 6, Lecture A: The Discrete Fourier Transform (DFT) Adam Hartz hz@mit.edu What is 6.003? What is a signal? Abstractly, a signal is a function that conveys information Signal
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More informationMultimedia Signals and Systems - Audio and Video. Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2
Multimedia Signals and Systems - Audio and Video Signal, Image, Video Processing Review-Introduction, MP3 and MPEG2 Kunio Takaya Electrical and Computer Engineering University of Saskatchewan December
More informationECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are
ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:
More informationDiscrete-time Signals and Systems in
Discrete-time Signals and Systems in the Frequency Domain Chapter 3, Sections 3.1-39 3.9 Chapter 4, Sections 4.8-4.9 Dr. Iyad Jafar Outline Introduction The Continuous-Time FourierTransform (CTFT) The
More informationChapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier
More informationLecture 8: Signal Reconstruction, DT vs CT Processing. 8.1 Reconstruction of a Band-limited Signal from its Samples
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 8: Signal Reconstruction, D vs C Processing Oct 24, 2001 Prof: J. Bilmes
More informationENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function
More information7.16 Discrete Fourier Transform
38 Signals, Systems, Transforms and Digital Signal Processing with MATLAB i.e. F ( e jω) = F [f[n]] is periodic with period 2π and its base period is given by Example 7.17 Let x[n] = 1. We have Π B (Ω)
More information!Sketch f(t) over one period. Show that the Fourier Series for f(t) is as given below. What is θ 1?
Second Year Engineering Mathematics Laboratory Michaelmas Term 998 -M L G Oldfield 30 September, 999 Exercise : Fourier Series & Transforms Revision 4 Answer all parts of Section A and B which are marked
More informationFourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides)
Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform How we will get there? Periodic discrete-time signal representation by Discrete-time
More informationECE 301. Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3
ECE 30 Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3 Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out
More informationFinal Exam of ECE301, Section 1 (Prof. Chih-Chun Wang) 1 3pm, Friday, December 13, 2016, EE 129.
Final Exam of ECE301, Section 1 (Prof. Chih-Chun Wang) 1 3pm, Friday, December 13, 2016, EE 129. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, and
More information3.2 Complex Sinusoids and Frequency Response of LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationHomework 6 EE235, Spring 2011
Homework 6 EE235, Spring 211 1. Fourier Series. Determine w and the non-zero Fourier series coefficients for the following functions: (a 2 cos(3πt + sin(1πt + π 3 w π e j3πt + e j3πt + 1 j2 [ej(1πt+ π
More informationFourier Representations of Signals & LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n] 2. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More informationECE-314 Fall 2012 Review Questions for Midterm Examination II
ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem
More informationSignals & Systems. Lecture 4 Fourier Series Properties & Discrete-Time Fourier Series. Alp Ertürk
Signals & Systems Lecture 4 Fourier Series Properties & Discrete-Time Fourier Series Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation:
More informationHomework 8 Solutions
EE264 Dec 3, 2004 Fall 04 05 HO#27 Problem Interpolation (5 points) Homework 8 Solutions 30 points total Ω = 2π/T f(t) = sin( Ω 0 t) T f (t) DAC ˆf(t) interpolated output In this problem I ll use the notation
More informationVU Signal and Image Processing. Torsten Möller + Hrvoje Bogunović + Raphael Sahann
052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/17s/
More informationDigital Signal Processing. Lecture Notes and Exam Questions DRAFT
Digital Signal Processing Lecture Notes and Exam Questions Convolution Sum January 31, 2006 Convolution Sum of Two Finite Sequences Consider convolution of h(n) and g(n) (M>N); y(n) = h(n), n =0... M 1
More informationIntroduction to DFT. Deployment of Telecommunication Infrastructures. Azadeh Faridi DTIC UPF, Spring 2009
Introduction to DFT Deployment of Telecommunication Infrastructures Azadeh Faridi DTIC UPF, Spring 2009 1 Review of Fourier Transform Many signals can be represented by a fourier integral of the following
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More informationTopic 3: Fourier Series (FS)
ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties
More informationThe Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002.
The Johns Hopkins University Department of Electrical and Computer Engineering 505.460 Introduction to Linear Systems Fall 2002 Final exam Name: You are allowed to use: 1. Table 3.1 (page 206) & Table
More informationDiscrete Fourier Transform
Discrete Fourier Transform Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz Diskrete Fourier transform (DFT) We have just one problem with DFS that needs to be solved.
More informationDSP Laboratory (EELE 4110) Lab#5 DTFS & DTFT
Islamic University of Gaza Faculty of Engineering Electrical Engineering Department EG.MOHAMMED ELASMER Spring-22 DSP Laboratory (EELE 4) Lab#5 DTFS & DTFT Discrete-Time Fourier Series (DTFS) The discrete-time
More informationQuestion Paper Code : AEC11T02
Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationEE 16B Final, December 13, Name: SID #:
EE 16B Final, December 13, 2016 Name: SID #: Important Instructions: Show your work. An answer without explanation is not acceptable and does not guarantee any credit. Only the front pages will be scanned
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationHomework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, Signals & Systems Sampling P1
Homework: 4.50 & 4.51 of the attachment Tutorial Problems: 7.41, 7.44, 7.47, 7.49 Signals & Systems Sampling P1 Undersampling & Aliasing Undersampling: insufficient sampling frequency ω s < 2ω M Perfect
More informationUniversity of Connecticut Lecture Notes for ME5507 Fall 2014 Engineering Analysis I Part III: Fourier Analysis
University of Connecticut Lecture Notes for ME557 Fall 24 Engineering Analysis I Part III: Fourier Analysis Xu Chen Assistant Professor United Technologies Engineering Build, Rm. 382 Department of Mechanical
More informationLecture 3 January 23
EE 123: Digital Signal Processing Spring 2007 Lecture 3 January 23 Lecturer: Prof. Anant Sahai Scribe: Dominic Antonelli 3.1 Outline These notes cover the following topics: Eigenvectors and Eigenvalues
More informationLaplace Transforms and use in Automatic Control
Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral
More informationThe Continuous-time Fourier
The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals:
More informationHomework 3 Solutions
EECS Signals & Systems University of California, Berkeley: Fall 7 Ramchandran September, 7 Homework 3 Solutions (Send your grades to ee.gsi@gmail.com. Check the course website for details) Review Problem
More informationFourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year
Fourier transform Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Methods for Image Processing academic year 27 28 Function transforms Sometimes, operating on a class of functions
More informationFourier Analysis and Spectral Representation of Signals
MIT 6.02 DRAFT Lecture Notes Last update: April 11, 2012 Comments, questions or bug reports? Please contact verghese at mit.edu CHAPTER 13 Fourier Analysis and Spectral Representation of Signals We have
More informationHomework 5 EE235, Summer 2013 Solution
Homework 5 EE235, Summer 23 Solution. Fourier Series. Determine w and the non-zero Fourier series coefficients for the following functions: (a f(t 2 cos(3πt + sin(πt + π 3 w π f(t e j3πt + e j3πt + j2
More informationECE 301: Signals and Systems Homework Assignment #3
ECE 31: Signals and Systems Homework Assignment #3 Due on October 14, 215 Professor: Aly El Gamal A: Xianglun Mao 1 Aly El Gamal ECE 31: Signals and Systems Homework Assignment #3 Problem 1 Problem 1 Consider
More informationChapter 3 Convolution Representation
Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn
More informationx[n] = x a (nt ) x a (t)e jωt dt while the discrete time signal x[n] has the discrete-time Fourier transform x[n]e jωn
Sampling Let x a (t) be a continuous time signal. The signal is sampled by taking the signal value at intervals of time T to get The signal x(t) has a Fourier transform x[n] = x a (nt ) X a (Ω) = x a (t)e
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 2-May-05 COURSE: ECE-2025 NAME: GT #: LAST, FIRST (ex: gtz123a) Recitation Section: Circle the date & time when
More informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More informationDigital Signal Processing Lecture 3 - Discrete-Time Systems
Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of
More information