Frequency Response of FIR Filters


 Beatrix Banks
 3 years ago
 Views:
Transcription
1 Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input of the form xn e jˆ n n. A fundamental result we shall soon see, is that the frequency response and impulse response are related through an operation known as the Fourier transform. The Fourier transform itself is not however formally studied in this chapter. Sinusoidal Response of FIR Systems Consider an FIR filter when the input is a complex sinusoid of the form xn Ae j e jˆ n n, (6.1) where it could be that xn was obtained by sampling the complex sinusoid xt Ae j e j t and ˆ T s From the difference equation for an M + 1tap FIR filter, M yn b k xn k b k Ae j e jˆ n k k M k (6.) ECE 61 Signal and Systems 61
2 Sinusoidal Response of FIR Systems yn Ae j e jˆ n k b k e j ˆ k Ae j e jˆ n jˆ He n where we have defined for arbitrary ˆ M (6.3) He jˆ M b k e j k ˆ k (6.4) to be the frequency response of the FIR filter Note: The notation He jˆ is used rather than say Hˆ, to be consistent with the ztransform which will defined in Chapter 7, and to emphasize the fact that the frequency response is periodic, with period (more on this later) Note: e jˆ cosˆ + jsinˆ, where sine and cosine are both mod functions Returning to (6.), the implication is that when the input is a complex exponential at frequency ˆ, the output is also a complex exponential at frequency ˆ The complex amplitude (magnitude and phase) of the input is changed as a result of passing through the system The frequency response at ˆ multiplies the input amplitude to produce the output It is not true in general that yn He jˆ xn, but only for the special case of xn a complex sinusoid applied starting at ECE 61 Signals and Systems 6
3 Sinusoidal Response of FIR Systems The frequency response is a complex function of ˆ generally viewed in either polar or rectangular form that is He jˆ He jˆ e j H ejˆ (6.5) ReHe jˆ + jimhe jˆ The polar or magnitude and phase form is perhaps the most common The polar form offers the following interpretation of terms of xn, when the input is a complex sinusoid yn e j H ejˆ He jˆ Ae j H ejˆ He jˆ yn in (6.6) Here we see that the input amplitude is multiplied by the frequency response amplitude, and the input phase has added to it the frequency response phase The output amplitude expression means that is also termed the gain of an LTI system Ae j e jˆ n + jˆ n e He jˆ Example: b k The frequency response of this FIR filter is b k e jˆ k He jˆ 4 k 1 + e jˆ + 3e jˆ + e j3ˆ + e j4ˆ ECE 61 Signals and Systems 63
4 Sinusoidal Response of FIR Systems We have used the inverse Euler formula for cosine twice For this particular filter we have that Why? e jˆ He jˆ He jˆ e jˆ e jˆ e jˆ e jˆ + e jˆ cosˆ + cosˆ cosˆ + cosˆ He jˆ ˆ Use MATLAB to plot the magnitude and phase response >> w :*pi/:*pi; >> H exp(j**w).*(3 + *cos(w) + *cos(*w)); >> subplot(11) >> plot(w,abs(h)) >> axis([ *pi 8]) >> grid >> ylabel('magnitude') >> subplot(1) >> plot(w,angle(h)) >> axis([ *pi pi pi]) >> grid >> ylabel('phase (rad)') ECE 61 Signals and Systems 64
5 Sinusoidal Response of FIR Systems >> xlabel('hat(\omega) (rad)') 8 Magnitude Phase (rad) hat(ω) (rad) Example: Find yn for Input The input frequency is the phase is xn 5e j1 n ˆ 1 rad, the amplitude is 5, and Assuming xn is input to the 4tap FIR filter in the previous example, the filter output is yn 3 + cos1 + cos 1e j e j e jn The amplitude response or gain at ˆ 1 is He j1 3.48; why? 5e j1 n ECE 61 Signals and Systems 65
6 Superposition and the Frequency Response Superposition and the Frequency Response We can use the linearity of the FIR filter to compute the output to a sum of sinusoids input signal As a special case we first consider a single real sinusoid xn Acosˆ n + (6.7) Using Euler s formula we expand (6.7) xn A  e j ˆ n + (6.8) The filter output due to each complex sinusoid is known from (6.3), so now using superposition we can write yn (6.9) We can simplify this result to a nice compact form, if we make the assumption that the FIR filter has real coefficients Special Result: It will be shown in a later section of this chapter that an FIR filter with real coefficients has conjugate symmetry What does this mean? A A  He jˆ j ˆ e n + He jˆ He jˆ A +  e j ˆ n + ˆ +  He j H* e jˆ e j ˆ n + ReHe jˆ jimhe jˆ He jˆ He jˆ (6.1) ECE 61 Signals and Systems 66
7 We now use (6.1) to simplify (6.9) yn A  He jˆ j ˆ e n + A Superposition and the Frequency Response  H* e jˆ j + e ˆ n + AH e jˆ e j ˆ n + + He jˆ e j ˆ n + +He jˆ (6.11) yn AH e jˆ cosˆ n + + He jˆ We see that when a real sinusoid passes through an LTI system, such as an FIR filter (having real coefficients), the output is also a real sinusoid which has picked up the magnitude and phase of the system at ˆ ˆ The generalization (sum of sinusoids) of this result is when N xn X + X k cosˆ kn + X k, (6.1) k 1 then the corresponding LTI system output is yn X He j + N k 1 X k He jˆ k cos ˆ kn+ X k +He jˆ k (6.13) ECE 61 Signals and Systems 67
8 Superposition and the Frequency Response Example: Three Inputs with The input is The frequency response is The input frequencies are He j He j 4 He j 3 Thus b k 1 11 xn n cos n cos e jˆ + e jˆ He jˆ e jˆ 1 + cosˆ >> w :pi/:pi; >> H 1  exp(j*w) + exp(j**w); >> subplot(11) >> plot(w,abs(h)) >> grid >> hold >> plot([pi/4 pi/4],[ 3],'r') ˆ k 4 3 e j 1 + cos 1 e j cos 4.414e j 4 e j cos 3 yn cos n cos n ECE 61 Signals and Systems 68
9 Superposition and the Frequency Response >> plot([pi/3 pi/3],[ 3],'r') >> ylabel('magnitude') >> subplot(1) >> plot(w,angle(h)) >> grid >> hold >> plot([pi/4 pi/4],[ 3],'r') >> ylabel('phase (rad)') >> xlabel('digital Frequency \omega') He jˆ 3.5 Magnitude Phase (rad) Digital Frequency ω We have used frequency domain analysis to complete this example We could also perform a time domain analysis using say MATLAB s filter function (more on this in the next section) ECE 61 Signals and Systems 69
10 SteadyState and Transient Response SteadyState and Transient Response The frequency response definition relies on the fact that the support interval for the complex sinusoid input is In a computer simulation or in a realtime signal processing application, it is not practical to consider input signals which begin at Consider an input that begins at n using the unit step function to turn on the input where X Ae j xn un and Xe jˆ n un 1, n, otherwise (6.14) (6.15) For an LTI FIR system, the convolution sum formula yields M yn b k Xe jˆ n k un k k hk (6.16)... M un k n k k ECE 61 Signals and Systems 61
11 The sum of (6.16) is composed of three cases When yn n, n n b k e jˆ k jˆ Xe n, n M k M b k e jˆ k jˆ n Xe, n M k SteadyState and Transient Response the input is not present, so the output is zero (6.17) When n M the input has partially engaged the filter and the output produced is the transient response When n M, the output is in steadystate, since the input has fully engaged the filter Note that the steadystate response is also equivalent to saying yn He jˆ Xe jˆ (6.18) Note also that the steadystate output assumes that the input is always Xe jˆ n for n Example: Revisit Three Inputs with n M b k 1 11 The frequency domain analysis used in the previous example obtained just the steadystate portion of the output We can evaluate the convolution sum or use MATLAB s filter function to obtain a solution of the form described in (6.17) ECE 61 Signals and Systems 611
12 SteadyState and Transient Response >> n 5:; >> x (1 + 4*cos(pi/4*n+pi/8) *cos(pi/3*npi/4)).*ustep(n,); >> y filter([11 1],1,x); >> subplot(11) >> stem(n,x,'filled') >> grid >> ylabel('x[n]') >> subplot(1) >> stem(n,y,'filled') >> hold Current plot released >> stem(n,y,'filled') >> grid >> ylabel('y[n]') >> xlabel('sample Index n') 15 x[n] Transient Interval y[n] 1 5 Steadystate Sample Index n ECE 61 Signals and Systems 61
13 The function ustep() is defined as: SteadyState and Transient Response function x ustep(n,n) % function x ustep(n,n) % % Generate a time shifted unit step sequence x zeros(size(n)); i_n find(n > n); x(i_n) ones(size(i_n)); If we consider just the steadystate portion of the output we should be able to discern the same output as obtained from the frequency domain analysis of the previous example y[n] DC 1 y[n] peak lag 1/ sample / x[n] cos() peak Sample Index n The measurements above agree with the earlier example ECE 61 Signals and Systems 613
14 Properties of the Frequency Response Properties of the Frequency Response Relation to Impulse Response and Difference Equation In the definition of the frequency response, He jˆ, for an FIR filter, the coefficients were utilized Similarly given the frequency response, the impulse response can be found hn In Chapter 5 we saw that the impulse response and difference equation are directly related, so in summary given one the other two are easy to obtain Example: Given say M we immediately see that The difference equation is The frequency response is b k Time Domain Frequency Domain hk n k He jˆ hk e jˆ k hn k hn yn He jˆ He jˆ M k n + 3n 1 n 3 b k xn + 3xn 1 xn ˆ + 3e j e jˆ ECE 61 Signals and Systems 614
15 Properties of the Frequency Response Example: Difference Equation from Suppose that We immediately can write that and It could have been that started in this form Example: hn yn He jˆ He jˆ We need to convert the sine form back to complex exponentials The impulse response is thus? ˆ He jˆ 1+ 4e j + He jˆ 4e j4ˆ n + 4n + 4n 4 xn + 4xn + 4xn 4 1 8e jˆ e jˆ cosˆ e jˆ j ˆ e jˆ sin je j ˆ He jˆ e jˆ 1 e jˆ e jˆ e jˆ j ECE 61 Signals and Systems 615
16 Properties of the Frequency Response The difference equation is? Periodicity of He jˆ For any discretetime LTI system, the frequency response is periodic, that is He j ˆ + He jˆ The proof for FIR filters follows He j ˆ M + b k e j ˆ + k M k b k e jˆ (6.19) (6.) This result is consistent with the fact that sinusoidal signals are insensitive to frequency shifts, i.e., In summary, the frequency response is unique on at most a interval, say in particular the interval ˆ Conjugate Symmetry The frequency response He jˆ is in general a complex quantity, in most cases obeys certain symmetry properties e j 1 He jˆ xn Xe j ˆ + n Xe jˆ n jn e Xe jˆ n ECE 61 Signals and Systems 616
17 Properties of the Frequency Response In particular if the coefficients of a digital filter are real, that * is b k b k or hk h * k, then proof Conjugate Symmetry: He jˆ H * e jˆ H * e jˆ b k e jˆ k * M * jˆ k b k e A consequence of conjugate symmetry is that (6.1) (6.) (6.3) We say that the magnitude response is an even function of ˆ and the phase is an odd function of ˆ It also follows that M k M k b k e j ˆ k k He jˆ When plotting the frequency response we can use the above symmetry to just plot over the interval ˆ He jˆ He jˆ He jˆ He j ˆ ReHe jˆ ReHe jˆ (6.4) ImHe jˆ ImHe jˆ We say that the real part response is an even function of ˆ and the imaginary part is an odd function of ˆ ECE 61 Signals and Systems 617
18 Graphical Representation of the Frequency Response Graphical Representation of the Frequency Response A useful MATLAB function for plotting the frequency response of any discretetime (digital) filter is freqz() The interface to freqz() is similar to filter() in that a and b vectors are again required Recall that the b vector holds the FIR coefficients and for FIR filters we set a 1. >> help freqz FREQZ Digital filter frequency response. [H,W] FREQZ(B,A,N) returns the Npoint complex frequency response vector H and the Npoint frequency vector W in radians/sample of the filter: jw jw jmw jw B(e) b(1) + b()e b(m+1)e H(e) jw jw jnw A(e) a(1) + a()e a(n+1)e given numerator and denominator coefficients in vectors B and A. The frequency response is evaluated at N points equally spaced around the upper half of the unit circle. If N isn't specified, it defaults to 51. [H,W] FREQZ(B,A,N,'whole') uses N points around the whole unit circle. H FREQZ(B,A,W) returns the frequency response at frequencies designated in vector W, in radians/sample (normally between and pi). [H,F] FREQZ(B,A,N,Fs) and [H,F] FREQZ(B,A,N,'whole',Fs) return frequency vector F (in Hz), where Fs is the sampling frequency (in Hz). H FREQZ(B,A,F,Fs) returns the complex frequency response at the frequencies designated in vector F (in Hz), where Fs is the sampling frequency (in Hz). FREQZ(B,A,...) with no output arguments plots the magnitude and unwrapped phase of the filter in the current figure window. See also filter, fft, invfreqz, fvtool, and freqs. b k ECE 61 Signals and Systems 618
19 Graphical Representation of the Frequency Response Example: Delay System yn xn n This filter contains one coefficient, b n 1, so He jˆ e jˆ n >> [H,w] freqz([ 1],1); >> subplot(11) >> plot(w,abs(h)) >> axis([ pi 1.]); grid >> ylabel('magniude Response') >> subplot(1) >> plot(w,angle(h)) >> axis([ pi pi pi]); grid >> ylabel('phase Response (rad)') >> xlabel('hat(\omega)') 1 ˆ n Magnitude Response 1.5 Constant magnitude (gain) response n Phase Response (rad) Linear Phase 4ˆ MATLAB wraps the phase mod hat(ω) ECE 61 Signals and Systems 619
20 Graphical Representation of the Frequency Response Example: FirstDifference System The frequency response is He jˆ yn xn xn 1 1 e jˆ 1 cosˆ + jsinˆ >> w pi:pi/1:pi; >> H freqz([11],1,w); %use a custom w axis >> subplot(11) >> plot(w,abs(h)) >> axis([pi pi ]); grid; ylabel('magniude Response') >> subplot(1) >> plot(w,angle(h)) >> axis([pi pi  ]); grid; >> ylabel('phase Response (rad)') >> xlabel('hat(\omega)') Magnitude Response sinˆ Phase Response (rad) 1 1 atan sinˆ cosˆ hat(ω) ECE 61 Signals and Systems 6
21 Graphical Representation of the Frequency Response Example: A Simple Lowpass Filter yn xn + xn 1 + xn The frequency response is He jˆ 1 e jˆ e jˆ >> w pi:pi/1:pi; >> H freqz([1 1],1,w); >> subplot(11) >> plot(w,abs(h)) >> axis([pi pi 4]); grid; >> ylabel('magnitude Response') >> subplot(1) >> plot(w,angle(h)) >> axis([pi pi pi pi]); grid; >> ylabel('phase Response (rad)') >> xlabel('hat(\omega)') + + e jˆ + cosˆ Magnitude Response cosˆ Phase Response (rad) ˆ hat(ω) ECE 61 Signals and Systems 61
22 Cascaded LTI Systems Cascaded LTI Systems Cascaded LTI system were discussed in Chapter 5 from the timedomain standpoint In the frequencydomain there is an alternative view of the input/output relationships To begin with consider again xn e jˆ n n With LTI #1 followed by LTI #, the output at frequency y 1 n H e jˆ H e jˆ H 1 e jˆ H 1 e jˆ e jˆ n e jˆ n (6.5) ˆ is (6.6) ECE 61 Signals and Systems 6
23 Cascaded LTI Systems With LTI # followed by LTI #1, the output at frequency y n H 1 e jˆ H 1 e jˆ H e jˆ H e jˆ e jˆ n e jˆ n is (6.7) Since H 1 e jˆ H e jˆ H e jˆ H 1 e jˆ, it follows that y 1 n y n, and we have established that the frequency response for a cascaded system is simply the product of the frequency responses ˆ H cascade e jˆ He jˆ H 1 e jˆ H e jˆ (6.8) We have also shown that a convolution in the timedomain is equivalent to a multiplication in the frequencydomain hn Example: Two System Cascade Consider Time Domain Frequency Domain h 1 n*h n H 1 e jˆ H e jˆ He jˆ H 1 e jˆ 1 + e jˆ H e jˆ 1+ e jˆ + The cascade frequency response is He jˆ e jˆ 1 + e jˆ 1+ e jˆ + 1 3e jˆ + + 3e jˆ + e jˆ e j3ˆ (6.9) ECE 61 Signals and Systems 63
24 Moving Average Filtering Given the cascade frequency response we can easily convert back to the impulse response of the cascade difference equation hn n + 3n 1+ 3n + n 3 and yn xn + 3xn 1 + 3xn + xn 3 Check h 1 n*h n Moving Average Filtering The moving average filter occurs frequency enough that we should consider finding a general expression for the frequency response The difference equation for an Lpoint averager is yn The frequency response is L xn k L k (6.3) He jˆ L e jˆ k L k (6.31) ECE 61 Signals and Systems 64
25 This sum is known as a finite geometric series It can be shown (discussed in class) that Moving Average Filtering L 1 k k 1 L , 1 L, 1 (why?) Apply (6.3) to (6.31) by setting e jˆ (6.3) He jˆ e jˆ L L 1 e jˆ  1 e jˆ L e jˆ L e jˆ L L e jˆ e jˆ e jˆ sinˆ L jˆ L 1 e Lsinˆ (6.33) Notice that the phase response is composed of a linear term e jˆ L 1 and due to the sign changes of sinˆ L / Lsinˆ In MATLAB D L e jˆ diricˆ L sinˆ L Lsinˆ can be used for analyzing moving average filters ECE 61 Signals and Systems 65
26 Moving Average Filtering Plotting the Frequency Response The frequency response can be plotted most easily using MATLAB s freqz() function Consider a 1point moving average >> w pi:pi/5:pi; >> H freqz(ones(1,1)/1,1,w); >> subplot(11) >> plot(w,abs(h)) >> grid; axis([pi pi 1]) >> ylabel('magnitude Response') >> subplot(1) >> plot(w,angle(h)) >> grid; axis([pi pi pi pi]) >> ylabel('phase Response (rad)') >> xlabel('hat(\omega)') Magnitude Response 1.5 L ˆ L Phase Response (rad) e j4.5 ˆ on this segment hat(ω) ECE 61 Signals and Systems 66
27 Filtering Sampled ContinuousTime Signals Filtering Sampled ContinuousTime Signals Consider the system model as shown below Suppose that xt t Xe jt We know that after sampling we have (6.34) xn xnt s Xe jnt s Xe jˆ n (6.35) The key relationship to connect the continuoustime and the discretetime quantities is ˆ T s  f f ˆ  ˆ f s T s ˆ T s (6.36) As long as the input analog frequency satisfies the sampling theorem, i.e., T s or f f s, the output of the discretetime system will be f s ˆ f s ECE 61 Signals and Systems 67
28 Filtering Sampled ContinuousTime Signals yn He jˆ Xe jˆ n (6.37) We can write yn in terms of the analog frequency variable via ˆ T s yn He jt s Xe jt sn (6.38) At the output of the DtoC converter we have the reconstructed output yt He jt s Xe jt He jf f s Xe j f f st (6.39) where f is the input analog sinusoid frequency (perhaps a better notation would be f ˆ ) Example: Lowpass Averager Consider a 5point moving average filter wrapped up between a CtoD and DtoC system We assume a sampling rate of 1 Hz and an input composed of two sinusoids xt cos1t+ 3cos3t Find the system frequency response in terms of the analog frequency variable f, and find the steadystate output yt We will use freqz() to obtain the frequency response >> w pi:pi/1:pi; >> H freqz(ones(1,5)/5,1,w); >> subplot(11) ECE 61 Signals and Systems 68
29 Filtering Sampled ContinuousTime Signals >> plot(w,abs(h)) >> axis([pi pi 1]); grid >> ylabel('magnitude Response') >> subplot(1) >> plot(w,angle(h)) >> axis([pi pi pi pi]); grid >> ylabel('phase Response (rad)') >> xlabel('hat(\omega)') Magnitude Response 1.5 Magnitude and Phase Plots of He jˆ Phase Response (rad) hat(ω) >> subplot(11) >> plot(w*1/(*pi),abs(h)) >> grid >> ylabel('magnitude Response') >> subplot(1) >> plot(w*1/(*pi),angle(h)) >> grid >> ylabel('phase Response (rad)') >> xlabel('f (Hz)') ECE 61 Signals and Systems 69
30 Filtering Sampled ContinuousTime Signals Magnitude Response 1.5 Magnitude and Phase Plots of He jf f s Phase Response (rad) f (Hz) The output yt will be of the same form as the input xt, except the sinusoids at 1 and 3 Hz need to have the filter frequency response applied Note 1 Hz and 3 Hz < 1/ 5 Hz (no aliasing) To properly apply the filter frequency response we need to convert the analog frequencies to the corresponding discretetime frequencies 1Hz Hz (6.4) ECE 61 Signals and Systems 63
31 The frequency response for average filter is Filtering Sampled ContinuousTime Signals in the general moving The frequency response at these two frequencies is He j. He j.6 (6.41) sin e j..647e j.4 5sin. (6.4) sin e j.6.47e j1. 5sin.6 >> diric(.*pi,5) ans.647 He jˆ >> diric(.6*pi,5) L 5 sin.5ˆ e jˆ sinˆ.47e j. ans .47 % also.47 at angle +/ pi The filter output yn is yn.647cos.n cos.n 1. + and the DtoC output is yt.647cos1t cos3t. (6.43) (6.44) ECE 61 Signals and Systems 631
32 Filtering Sampled ContinuousTime Signals Interpretation of Delay In the study of the moving average filter, it was noted that where D L e jˆ He jˆ The pure delay system has impulse response and has frequency response D L e jˆ e jˆ L 1 is a purely real function yn xn n hn n n e jn ˆ He jˆ We also know that the impulse response of two systems in cascade is hn He jˆ h 1 n*h n The Lpoint moving average filter thus incorporates a delay of L 1 samples Viewed as a cascade of subsystems He jˆ If L is odd then the delay is an integer H 1 e jˆ H e jˆ D L e jˆ e jˆ L 1 If L is even then the delay is an odd half integer (6.45) ECE 61 Signals and Systems 63
SGN1158 Introduction to Signal Processing Test. Solutions
SGN1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2sided sheet of notes. No books, no other
More informationENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter
ENEE 425 Spring 2010 Assigned Homework Oppenheim and Shafer (3rd ed.) Instructor: S.A. Tretter Note: The dates shown are when the problems were assigned. Homework will be discussed in class at the beginning
More informationExercises in Signals, Systems, and Transforms
Exercises in Signals, Systems, and Transforms Ivan W. Selesnick Last edit: October 7, 4 Contents DiscreteTime Signals and Systems 3. Signals...................................................... 3. System
More information(Refer Slide Time: 01:1101:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture  6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
More informationEE 422G  Signals and Systems Laboratory
EE 4G  Signals and Systems Laboratory Lab 4 IIR Filters written by Kevin D. Donohue Department of Electrical and Computer Engineering University of Kentucky Lexington, KY 40506 September 6, 05 Objectives:
More informationUNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals
UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101  Fall 2010 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS (YOURS!) You are allowed one 2sided sheet of
More informationchapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective
Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything
More informationFFT Algorithms. Chapter 6. Contents 6.1
Chapter 6 FFT Algorithms Contents Efficient computation of the DFT............................................ 6.2 Applications of FFT................................................... 6.6 Computing DFT
More informationLab 4 Sampling, Aliasing, FIR Filtering
47 Lab 4 Sampling, Aliasing, FIR Filtering This is a software lab. In your report, please include all Matlab code, numerical results, plots, and your explanations of the theoretical questions. The due
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More informationThe Fourier Analysis Tool in Microsoft Excel
The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier
More informationFrequency Response MUMT 501 Digital Audio Signal Processing
Frequency Response MUMT 501 Digital Audio Signal Processing Gerald Lemay McGill University MUMT 501 Frequency Response 1 Table of Contents I Introduction... 3 I.1 Useful Math and Systems Concepts... 3
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationFast Fourier Transforms and Power Spectra in LabVIEW
Application Note 4 Introduction Fast Fourier Transforms and Power Spectra in LabVIEW K. Fahy, E. Pérez Ph.D. The Fourier transform is one of the most powerful signal analysis tools, applicable to a wide
More informationUniversity of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems. FFT Tutorial
University of Rhode Island Department of Electrical and Computer Engineering ELE 436: Communication Systems FFT Tutorial 1 Getting to Know the FFT What is the FFT? FFT = Fast Fourier Transform. The FFT
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya wwwsigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as
More informationDiscreteTime Signals and Systems
2 DiscreteTime Signals and Systems 2.0 INTRODUCTION The term signal is generally applied to something that conveys information. Signals may, for example, convey information about the state or behavior
More informationTCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS
TCOM 370 NOTES 994 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of
More informationController Design in Frequency Domain
ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion 1. Abstract
More informationFinal Year Project Progress Report. FrequencyDomain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones
Final Year Project Progress Report FrequencyDomain Adaptive Filtering Myles Friel 01510401 Supervisor: Dr.Edward Jones Abstract The Final Year Project is an important part of the final year of the Electronic
More informationSIGNAL PROCESSING & SIMULATION NEWSLETTER
1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty
More informationReview of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties
ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to
More informationThere are four common ways of finding the inverse ztransform:
Inverse ztransforms and Difference Equations Preliminaries We have seen that given any signal x[n], the twosided ztransform is given by n x[n]z n and X(z) converges in a region of the complex plane
More informationPolynomials and the Fast Fourier Transform (FFT) Battle Plan
Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and pointvalue representation
More informationCIRCUITS LABORATORY EXPERIMENT 3. AC Circuit Analysis
CIRCUITS LABORATORY EXPERIMENT 3 AC Circuit Analysis 3.1 Introduction The steadystate behavior of circuits energized by sinusoidal sources is an important area of study for several reasons. First, the
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationThe Z transform (3) 1
The Z transform (3) 1 Today Analysis of stability and causality of LTI systems in the Z domain The inverse Z Transform Section 3.3 (read class notes first) Examples 3.9, 3.11 Properties of the Z Transform
More informationDifference Equations
Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it
More informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Properties of the Fourier Transform Properties of the Fourier
More informationTTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT40 Digital Signal Processing Suggested Solution to Exam Fall 008 Problem (a) The input and the inputoutput
More information9 Fourier Transform Properties
9 Fourier Transform Properties The Fourier transform is a major cornerstone in the analysis and representation of signals and linear, timeinvariant systems, and its elegance and importance cannot be overemphasized.
More informationIntroduction to the Ztransform
ECE 3640 Lecture 2 Z Transforms Objective: Ztransforms are to difference equations what Laplace transforms are to differential equations. In this lecture we are introduced to Ztransforms, their inverses,
More informationComplex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers
Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension
More informationCHAPTER 6 Frequency Response, Bode Plots, and Resonance
ELECTRICAL CHAPTER 6 Frequency Response, Bode Plots, and Resonance 1. State the fundamental concepts of Fourier analysis. 2. Determine the output of a filter for a given input consisting of sinusoidal
More informationAnalysis/resynthesis with the short time Fourier transform
Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TUBerlin IRCAM Analysis/Synthesis
More informationIMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL
IMPLEMENTATION OF FIR FILTER USING EFFICIENT WINDOW FUNCTION AND ITS APPLICATION IN FILTERING A SPEECH SIGNAL Saurabh Singh Rajput, Dr.S.S. Bhadauria Department of Electronics, Madhav Institute of Technology
More informationANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 21
WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's
More informationConvolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005
Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall
More informationFrequency Response and Continuoustime Fourier Transform
Frequency Response and Continuoustime Fourier Transform Goals Signals and Systems in the FDpart II I. (Finiteenergy) signals in the Frequency Domain  The Fourier Transform of a signal  Classification
More informationLab 1. The Fourier Transform
Lab 1. The Fourier Transform Introduction In the Communication Labs you will be given the opportunity to apply the theory learned in Communication Systems. Since this is your first time to work in the
More informationmin ǫ = E{e 2 [n]}. (11.2)
C H A P T E R 11 Wiener Filtering INTRODUCTION In this chapter we will consider the use of LTI systems in order to perform minimum meansquareerror (MMSE) estimation of a WSS random process of interest,
More information1.1 DiscreteTime Fourier Transform
1.1 DiscreteTime Fourier Transform The discretetime Fourier transform has essentially the same properties as the continuoustime Fourier transform, and these properties play parallel roles in continuous
More informationThe Algorithms of Speech Recognition, Programming and Simulating in MATLAB
FACULTY OF ENGINEERING AND SUSTAINABLE DEVELOPMENT. The Algorithms of Speech Recognition, Programming and Simulating in MATLAB Tingxiao Yang January 2012 Bachelor s Thesis in Electronics Bachelor s Program
More informationPYKC Jan710. Lecture 1 Slide 1
Aims and Objectives E 2.5 Signals & Linear Systems Peter Cheung Department of Electrical & Electronic Engineering Imperial College London! By the end of the course, you would have understood: Basic signal
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More information3 Signals and Systems: Part II
3 Signals and Systems: Part II Recommended Problems P3.1 Sketch each of the following signals. (a) x[n] = b[n] + 3[n  3] (b) x[n] = u[n]  u[n  5] (c) x[n] = 6[n] + 1n + (i)2 [n  2] + (i)ag[n  3] (d)
More informationOriginal Lecture Notes developed by
Introduction to ADSL Modems Original Lecture Notes developed by Prof. Brian L. Evans Dept. of Electrical and Comp. Eng. The University of Texas at Austin http://signal.ece.utexas.edu Outline Broadband
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationSignal Processing First Lab 01: Introduction to MATLAB. 3. Learn a little about advanced programming techniques for MATLAB, i.e., vectorization.
Signal Processing First Lab 01: Introduction to MATLAB PreLab and WarmUp: You should read at least the PreLab and Warmup sections of this lab assignment and go over all exercises in the PreLab section
More informationFrequency response of a general purpose singlesided OpAmp amplifier
Frequency response of a general purpose singlesided OpAmp amplifier One configuration for a general purpose amplifier using an operational amplifier is the following. The circuit is characterized by:
More informationCHAPTER 10 Linear TimeInvariant (LTI) Models for Communication Channels
MIT 6.02 DRAFT Lecture Notes Fall 2011 (Last update: November 5, 2011) Comments, questions or bug reports? Please contact verghese at mit.edu CHAPTER 10 Linear TimeInvariant (LTI) Models for Communication
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationPurpose of Time Series Analysis. Autocovariance Function. Autocorrelation Function. Part 3: Time Series I
Part 3: Time Series I Purpose of Time Series Analysis (Figure from Panofsky and Brier 1968) Autocorrelation Function Harmonic Analysis Spectrum Analysis Data Window Significance Tests Some major purposes
More informationMATHEMATICAL SURVEY AND APPLICATION OF THE CROSSAMBIGUITY FUNCTION
MATHEMATICAL SURVEY AND APPLICATION OF THE CROSSAMBIGUITY FUNCTION Dennis Vandenberg Department of Mathematical Sciences Indiana University South Bend Email Address: djvanden@iusb.edu Submitted to the
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationDTFT, Filter Design, Inverse Systems
EE3054 Signals and Systems DTFT, Filter Design, Inverse Systems Yao Wang Polytechnic University Discrete Time Fourier Transform Recall h[n] H(e^jw) = H(z) z=e^jw Can be applied to any discrete time
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationAnalog and Digital Signals, Time and Frequency Representation of Signals
1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals
More informationDigital Signal Processing IIR Filter Design via Impulse Invariance
Digital Signal Processing IIR Filter Design via Impulse Invariance D. Richard Brown III D. Richard Brown III 1 / 11 Basic Procedure We assume here that we ve already decided to use an IIR filter. The basic
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationSECTION 6 DIGITAL FILTERS
SECTION 6 DIGITAL FILTERS Finite Impulse Response (FIR) Filters Infinite Impulse Response (IIR) Filters Multirate Filters Adaptive Filters 6.a 6.b SECTION 6 DIGITAL FILTERS Walt Kester INTRODUCTION Digital
More informationIntroduction to IQdemodulation of RFdata
Introduction to IQdemodulation of RFdata by Johan Kirkhorn, IFBT, NTNU September 15, 1999 Table of Contents 1 INTRODUCTION...3 1.1 Abstract...3 1.2 Definitions/Abbreviations/Nomenclature...3 1.3 Referenced
More informationClass Note for Signals and Systems. Stanley Chan University of California, San Diego
Class Note for Signals and Systems Stanley Chan University of California, San Diego 2 Acknowledgement This class note is prepared for ECE 101: Linear Systems Fundamentals at the University of California,
More informationDesign of Efficient Digital Interpolation Filters for Integer Upsampling. Daniel B. Turek
Design of Efficient Digital Interpolation Filters for Integer Upsampling by Daniel B. Turek Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements
More informationFrequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao
Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Timedomain description Waveform representation
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More information7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )
34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationDepartment of Electrical and Computer Engineering BenGurion University of the Negev. LAB 1  Introduction to USRP
Department of Electrical and Computer Engineering BenGurion University of the Negev LAB 1  Introduction to USRP  11 Introduction In this lab you will use software reconfigurable RF hardware from National
More informationSAMPLE SOLUTIONS DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou
SAMPLE SOLUTIONS DIGITAL SIGNAL PROCESSING: Signals, Systems, and Filters Andreas Antoniou (Revision date: February 7, 7) SA. A periodic signal can be represented by the equation x(t) k A k sin(ω k t +
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 22 Introduction to Fourier Transforms Fourier transform as a limit
More informationEquivalent Circuits and Transfer Functions
R eq isc Equialent Circuits and Transfer Functions Samantha R Summerson 14 September, 009 1 Equialent Circuits eq ± Figure 1: Théenin equialent circuit. i sc R eq oc Figure : MayerNorton equialent circuit.
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MT OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 00 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More information2 Signals and Systems: Part I
2 Signals and Systems: Part I In this lecture, we consider a number of basic signals that will be important building blocks later in the course. Specifically, we discuss both continuoustime and discretetime
More informationFUNDAMENTALS OF ENGINEERING (FE) EXAMINATION REVIEW
FE: Electric Circuits C.A. Gross EE11 FUNDAMENTALS OF ENGINEERING (FE) EXAMINATION REIEW ELECTRICAL ENGINEERING Charles A. Gross, Professor Emeritus Electrical and Comp Engineering Auburn University Broun
More informationChapter 8  Power Density Spectrum
EE385 Class Notes 8/8/03 John Stensby Chapter 8  Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationSampling and Interpolation. Yao Wang Polytechnic University, Brooklyn, NY11201
Sampling and Interpolation Yao Wang Polytechnic University, Brooklyn, NY1121 http://eeweb.poly.edu/~yao Outline Basics of sampling and quantization A/D and D/A converters Sampling Nyquist sampling theorem
More information1.4 Fast Fourier Transform (FFT) Algorithm
74 CHAPTER AALYSIS OF DISCRETETIME LIEAR TIMEIVARIAT SYSTEMS 4 Fast Fourier Transform (FFT Algorithm Fast Fourier Transform, or FFT, is any algorithm for computing the point DFT with a computational
More informationThe ztransform. The Nature of the zdomain. X(s) '
CHAPTER 33 The ztransform Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the ztransform. The overall strategy
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationSignals and Systems. Richard Baraniuk NONRETURNABLE NO REFUNDS, EXCHANGES, OR CREDIT ON ANY COURSE PACKETS
Signals and Systems Richard Baraniuk NONRETURNABLE NO REFUNDS, EXCHANGES, OR CREDIT ON ANY COURSE PACKETS Signals and Systems Course Authors: Richard Baraniuk Contributing Authors: Thanos Antoulas Richard
More informationL9: Cepstral analysis
L9: Cepstral analysis The cepstrum Homomorphic filtering The cepstrum and voicing/pitch detection Linear prediction cepstral coefficients Mel frequency cepstral coefficients This lecture is based on [Taylor,
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationFrequency Domain and Fourier Transforms
Chapter 4 Frequency Domain and Fourier Transforms Frequency domain analysis and Fourier transforms are a cornerstone of signal and system analysis. These ideas are also one of the conceptual pillars within
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationTrigonometry Lesson Objectives
Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the
More informationMethods for Vibration Analysis
. 17 Methods for Vibration Analysis 17 1 Chapter 17: METHODS FOR VIBRATION ANALYSIS 17 2 17.1 PROBLEM CLASSIFICATION According to S. H. Krandall (1956), engineering problems can be classified into three
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationMoving Average Filters
CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average
More informationCourse overview Processamento de sinais 2009/10 LEA
Course overview Processamento de sinais 2009/10 LEA João Pedro Gomes jpg@isr.ist.utl.pt Instituto Superior Técnico Processamento de sinais MEAer (IST) Course overview 1 / 19 Course overview Motivation:
More informationBharathwaj Muthuswamy EE100 Active Filters
Bharathwaj Muthuswamy EE100 mbharat@cory.eecs.berkeley.edu 1. Introduction Active Filters In this chapter, we will deal with active filter circuits. Why even bother with active filters? Answer: Audio.
More informationEE 221 AC Circuit Power Analysis. Instantaneous and average power RMS value Apparent power and power factor Complex power
EE 1 AC Circuit Power Analysis Instantaneous and average power RMS value Apparent power and power factor Complex power Instantaneous Power Product of timedomain voltage and timedomain current p(t) =
More informationIntroduction to Frequency Domain Processing 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Introduction to Frequency Domain Processing 1 1 Introduction  Superposition In this
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationNyquist Sampling Theorem. By: Arnold Evia
Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier
More informationIntroduction to Complex Fourier Series
Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function
More information