Sinusoidal signal. Arbitrary signal. Periodic rectangular pulse. Sampling function. Sampled sinusoidal signal. Sampled arbitrary signal
|
|
- Dinah Barrett
- 5 years ago
- Views:
Transcription
1 Techniques o Physics Worksheet 4 Digital Signal Processing 1 Introduction to Digital Signal Processing The ield o digital signal processing (DSP) is concerned with the processing o signals that have been converted into digital data. DSP techniques have ound application in many areas including image processing, instrumentation and control, data compression, audio, telecommunications and biomedicine to name but a ew. Although primarily a subject area o electronic engineering, an understanding o DSP can be extremely valuable to anyone involved in science and technology, including physicists. Modern physics experiments invariably use a data acquisition (DAQ) system to collect and record data in digital orm. The methods o DSP are then used to extract the required inormation rom the data. This might be done in real-time as the data is acquired or `o-line' as the irst step in the analysis o the data. 1.1 Discrete-time Systems A signal processor can be pictured as a `black box' with an input that receives a signal and an output that transmits a version o that signal ater it has been transormed in some way. A digital signal processor receives and processes discrete-time signals i.e. signals that have been sampled at regular time intervals 1. A sampled signal is basically a representation o a continuous signal d(t) by its values at discrete instants o time, x n = d(nt ), where T is the interval between regular samples, and n =; 1; 2;:::. Sampled signals are usually digitised, i.e. each sample is converted to a numerical value representing the magnitude o the signal at the instant o the sample, which enables them to be processed digitally. With both sampling and digitisation there is a loss o inormation and there are resulting side eects. An understanding o the side eects is essential when considering DSP systems. A system that process discrete-time signals is said to be linear i it obeys the principle o superposition, i.e. the response o the system to two or more inputs is equal to the sum o the separate responses to each input in the absence o other inputs. For example, i a discrete-time input signal a n gives rise to the output signal c n and b n gives rise to d n then the input signal a n + b n produces c n + d n. A discrete-time system is said to be 1 Signals need not be time-based but or the purposes o this document we are going to always talk in terms o time-based signals 1
2 } Figure 1: The convolution o a sampled signal with a real-time system. time invariant i its output does not depend on the time that the input is applied, i.e. the input signal x n gives output y n then the time-shited input signal x n+i will give the output y n+i. When a discrete-time system is both linear and time invariant its output y n or an input signal x n is given by the convolution sum y n = 1X k= h k x n k (1) where h k is the impulse response o the system. The convolution operation is shown graphically in igure 1; each output is generated by a sum o products between one input x n k and one coeicient rom the impulse response o the discrete-time system, h k. As a side remark, one might expect the convolution to be symmetrically centred around a given x n value, using input data rom both x n+i and x n i. However, in a real-time data processing system, where one wants to generate an output with minimum delay ater acquiring each new input sample, the x n+i data are not available at the time o the acquisition o x n, and so the convolution sum must work backwards rom the most recently acquired data. I the input signal is in the orm o an impulse i.e. x n = :::;; ; 1; ; ;:::the output o the system is h k (although shited in time.) Any input signal can be considered as a sequence o impulses with dierent amplitudes so, by the principle o superposition, the output is a sum o the responses to the individual input impulses. Hence, the output is the convolution o the input signal with impulse response described by equation 1. It is important to appreciate that the values o impulse response completely deine the system. Once you know the impulse response you know everything about the system and can calculate the output or any input signal. An alternative and complementary way o looking at signals and the response o systems is in the requency domain. Signals can be described by the amplitude and phase o their requency components and systems can be described by their requency response. 2
3 Converting between time domain and requency domain descriptions involves the Fourier transorm, which allows a periodic signal d(t) with period T to be decomposed as a set o components: d(t) = C n = 1 T 1X n= 1 Z T=2 C n e j2ßnt=t = T=2 1X n= 1 C n (cos 2ßnt=T + j sin 2ßnt=T ) d(t) e j2ßnt=t dt (2) A time domain signal x(t) and its requency domain equivalent X(!) orm a Fourier transorm pair and are related by: X(!) = F [x(t)] x(t) = F 1 [X(!)] (3) where F represents the Fourier transorm and F 1 represents the inverse Fourier transorm. In general, X(!) is complex and can be written in the amplitude-phase orm jx(!)j exp(i X (!)) where X (!) is the requency dependent phase angle. Similarly, a system's impulse response h(t) orms a Fourier transorm pair with the system's requency response or transer unction, H(!): H(!) = F(h(t)) h(t) = F 1 (H(!)): (4) Since H(!) is a complex unction it can modiy both the amplitude spectrum and the phase spectrum o the input signal. In the requency domain, the output o a system, Y (!), is calculated by multiplying the input, X(!), by the transer unction H(!) i.e. Y (!) =H(!)X(!): (5) Equation 5 is the requency domain equivalent o convoluting the input signal in the time domain with the impulse response, as can be seen by taking the Fourier transorm o y(t) = h(t) Λ x(t) where the Λ operator represents convolution: F [y(t)] = F [h(t) Λ x(t)] F [y(t)] = F [h(t)] F [x(t)] Y (!) = H(!)X(!): (6) In this working we have made use o the convolution theorem which states that convolution in the time domain is equivalent to multiplication in the requency domain.e. F [x 1 (t) Λ x 2 (t)] = X 1 (!)X 2 (!): (7) 3
4 Similarly, multiplication in the time domain is equivalent to convolution in the requency domain i.e. F [x 1 (t)x 2 (t)] = X 1 (!) Λ X 2 (!): (8) When thinking about signal processing problems it can be very helpul to keep these two relationships in mind. Figures 2a) d) show a number o kinds o signals in the time domain, and their representation in the requency domain, which can be obtained by the application o the Fourier Transorm. The requency domain diagrams show only the magnitude o the requency components, without any phase inormation; in general, the C n coeicient o any component is complex. A pure sine wave has only one requency component, so appears in a) as a delta unction at the requency o the signal. Note that a corresponding requency component exists with negative requency, but the coeicient is o opposite sign. The more general signal shown in b) is made up o a range o requency components, and in the arbitrary (non-periodic) signal case these components will orm a continuous distribution. For a periodic signal, such as the rectangular pulse shown in c), then only discrete requency components are required. The components are spaced by 1=T, where T is the period o the signal, and or the rectangular wave the coeicients o the requency components are given by the sinc() unction, with nodes spaced by 1=i, where tau is the width o each pulse. A special case o the periodic rectangular pulse is the sampling unction shown in d), where the width o the pulse has been reduced to zero. The components are still spaced by 1=T, where T is now the sampling period, and hence s =1=T is the sampling requency. The components are all o uniorm height since the nodes o the envelope are spaced at ±n1 realistic sampling involves non-zero sampling times, and so the distribution o components is not completely uniorm. 1.2 Sampling A sampled signal can be considered as being non-zero only at the regular sampling instants and zero at other times. In the time domain, this is equivalent to multiplying the continuous analogue signal by the ininite sequence o Dirac delta unctions (spaced by the sampling period T ) which make up the sampling unction. As has already been stated in Section 1.1, multiplying in the time domain is equivalent to convolution in the requency domain so we need to take the Fourier transorms o the input signal and the sampling unction and convolute them. First, consider again the pure sine wave; when multiplied in the time domain with the sampling unction, one obtains the sampled signal shown in Figure 2.e). The convolution o the delta unction o the signal with the sampling unction results in the signal appearing with an oset o d on either side o each integer multiple o the sampling requency, s. This pattern o signal appearing in upper and lower sidebands around repeated harmonics o the sampling requency, s, can be understood by expanding the 4
5 a) Time Domain Frequency Domain Fourier Transorm Sinusoidal signal t b) Arbitrary signal d t c) T Periodic rectangular pulse 1/T d) T t -3/ -2/ -1/ Sampling unction 1/ 2/ 3/ t -s s=1/t e) Multiplication Convolution Sampled sinusoidal signal ) t -s-d -s+d -d d Sampled arbitrary signal s-d s+d t Figure 2: Time-domain Frequency-domain equivalence and the eect o sampling a signal. 5 -s s
6 sampling unction, consisting o delta unctions at times t = ;T;2T :::, as a Fourier series in terms o its cosine components as z(t) =a + a 1 cos! s t + a 2 cos 2! s t + (9) where! s =2ß=T. I d(t) is a sinusoidal signal, sin! d t, then the sampled waveorm x(t) is given by multiplying the two time-domain unctions x(t) = sin! d t (a + a 1 cos! s t + a 2 cos 2! s t + ) = a sin! d t + a 1 2 sin(! s! d )t + a 1 2 sin(! s +! d )t + a 2 2 sin(2! s! d )t + a 2 2 sin(2! s +! d )t + (1) where the n! s ±! d terms ollow rom sin A cos B = [sin(a + B) + sin(a B)]=2. In general the input signal will consist o a range o requencies with an upper limit, which is perhaps determined by the limited requency response o a transducer (see Figure 2.b). When this signal is multiplied in the time domain by the sampling unction the result is the sampled signal shown in Figure 2.). The convolution o the requency domain representations o the input signal and the sampling unction is achieved by the same expansion around harmonics o s applied to every requency component o d(t). This results in the requency domain picture shown, with symmetric images o the input signal around each integer multiple o the sampling requency (only a ew are shown on the igure). Problems occur i the input signal contains requencies greater than s=2, the so-called Nyquist requency. In this case, the repeated spectra start to overlap as illustrated in Figure 3 and it becomes impossible to distinguish in this overlap between input requencies greater than s=2 and those less than s=2. This eect is called aliasing 2. The conclusion o this analysis is that requencies greater than s=2 cannot be recovered once the signal has been sampled at s. Not only is inormation rom requencies greater than s=2 lost, but these requencies appear as requencies below s=2, distorting the description o the requencies which are truly below s=2. To avoid the conusion o lower and higher requencies, it is good practice to remove the higher requencies by applying a low-pass ilter at the input o sampling circuit to remove all requency components above s=2. I the signal contains useul inormation at requencies > s=2 then it is necessary to increase the sampling requency. 1.3 The Fast Fourier Transorm The ast Fourier transorm (FFT) is a very useul tool or estimating the requency content o signals. The algorithm is highly eicient and using modern microprocessors 2 The amiliar eect in ilms o spoked wheels appearing to rotate more slowly than we know tobe the case, or even in reverse, is closely related to aliasing; the rotation o the wheel corresponds to a periodic waveorm and the sampling is provided by the camera shutter. 6
7 -s s Figure 3: Aliasing in the requency domain. it is possible to do real-time requency analysis o signals in the audio requency range. The FFT is a special orm o the more general discrete Fourier transorm (DFT). In turn, the DFT is a discrete-time orm o the Fourier transorm. That is, the input time domain data is in the orm o a sequence o discrete values and the output is a set o discrete requency amplitude-phase values. The DFT and Fourier transorm are thereore related but are not exactly equivalent. The DFT o a sequence o N discrete-time values x n, where n = ; 1;:::N 1, is given by: X k = p 1 N 1 X x n e i2ßkn=n (11) N n= where k =; 1;:::N 1. Hence, the DFT returns N complex values X k which represent the amplitude and phase or the harmonic requencies = ks=n where s is the sampling requency. The inverse DFT is given by: X x n = p 1 N 1 X k e i2ßkn=n : (12) N k= The FFT is mathematically identical to the DFT but the number o data points is restricted to 2 M where M is an integer. FFT's are much aster because the algorithm takes advantage o computational redundancies in the DFT. The discrete nature o the DFT results in side eects which need to be appreciated when using the DFT (or FFT) to analyse signals. The irst problem is aliasing has already been discussed. This problem can be solved by increasing the sampling requency until the requencies o interest are below the Nyquist requency. The second problem occurs because a signal component with requency not exactly equal to one o the harmonic requencies = ks=n cannot be properly represented. The result is that its amplitude is shared between nearby harmonics. This eect can be reduced by increasing the number o data points either by analysing more points or, i that is not possible, by adding zero values to the end o the data. This improves the 7
8 No window - sine sine Rectangular window - sine sine Figure 4: The eect o a rectangular window on a sine wave in the requency domain. spectral resolution o the DFT by reducing the spacing o harmonic requencies,, since or an N-point DFT = s=n. The third problem is spectral leakage and is the result o analysing only the inite time interval N=s. In order to resolve a signal into a inite set o discrete requency components, the DFT assumes that the signal is periodic (non-periodic signals require a continuous spectrum o ininitessimally-spaced components). However, in obtaining a inite number (2 M ) o input samples a signal o ininite duration has eectively been multiplied by a rectangular window unction to give a signal o inite duration. In the requency domain this is equivalent toconvoluting the requency spectrum o the signal with the Fourier transorm o the window unction. The simple rectangular window unction in the time domain transorms to a sinc() unction in the requency domain, with the width and spacing o the sinc unction's lobes being inversely related to the width o the rectangular window. Convolution with the input signal, as shown in Figure 4, with a wide central lobe and the long tails o the sinc unction results in the smearing o each requency component in the signal, so that it leaks" across several requency bands. For requencies or which the sampling duration (NT) is an integer multiple o the true period o the signal, the assumption by the DFT o a periodic signal is valid and the DFT works as i a inite window unction had never been applied, hence no spectral leakage. The spectral leakage problem problem can be improved irstly by increasing the width o the time 8
9 window, and secondly by using window unctions other than a rectangle which shape the input waveorm to look more like a periodic waveorm. Such window unctions have shorter tails in the requency domain, and so introduce less spreading over neighbouring requencies when convoluted with the signal. 1.4 Digital Filters Digital ilters are discrete-time systems that modiy the amplitude and/or phase o signals in a requency-dependent way. Filters are usually used to extract only the requencies o interest rom a signal. Very oten ilters are used to remove noise which contaminates a signal. The great advantage o digital iltering over using analogue ilters is that virtually any kind o digital ilter can be realized and implemented in a lexible and convenient way. The properties o a ilter are determined completely by its impulse response and digital ilters can be designed to have any arbitrary impulse response. I the impulse response is o inite length the ilter is a Finite Impulse Response (FIR) ilter and the output can be calculated directly rom the convolution sum presented earlier. In the context o implementing a digital ilter, the values o the impulse response become the ilter coeicients. Designing FIR ilters is relatively straightorward. The impulse response required to implement the ilter is obtained as a Fourier Transorm o the desired requency response. A simple and useul example is the ideal low-pass ilter response H D (!) shown in Figure 5. A ilter with this ideal response removes all requencies above the cut-o angular requency! c (the requency range has been normalised so that the sampling angular requency is 2ß). Notice that the requency response repeats because o the discrete-time nature o the signals. The impulse response can be calculated by taking the inverse Fourier transorm o H D (!), resulting in h n =! c ß =! c ß sin(n! c ) n! c ; n 6= n = (13) where n is an integer and 1 <n<+1. Immediately we see that we have a problem because an ininite number o values are required. To produce a practical ilter it is necessary to use only a `window' o values o h n around n =. This is equivalent to multiplying the impulse response by a rectangular window and the result is that the requency response deviates rom the ideal low-pass response (in a manner analogous to the spectral leakage caused by multiplying a time-domain signal by a rectangular window). Rather than multiply the requency response by a rectangular window, other window unctions can be used to reduce some o these problems in the same way that spectra rom the FFT can be improved by using a suitable time-domain window unction. 9
10 H D (ω) -2π -ω c ω c 2π ω (normalised) Figure 5: Ideal low-pass ilter requency response. It is highly recommended that you do some supplementary reading on digital signal processing beore attempting this worksheet. Search or keywords Digital Signal Processing" or Digital Filters". Some example titles are listed at the end o the worksheet (any one o these should contain useul analyses o sampling and iltering and there are many similar texts). 2 Exercises Week 5, Session 1/2 2.1 Sampling Use Mathcad to simulate sampling o a sine wave o requency at a sampling requency s = 1 Hz. Use the orm sin(2ßnt) where initially =1Hz and T =1=s and plot the sample values on a graph 3. Note that sampled input data is best stored in an array o samples, rather than as a unction o time or sample number; this approach will be needed or the later exercises (it is also advisable to treat time in the same way, as t n = n t, and not to use integer values o time t = n). Describe the appearance o the sampled signal as you increase in several steps rom a ew Hz to the Nyquist requency s=2 and then above the Nyquist requency (make sure you include requencies just a ew Hz either side o the Nyquist requency). At exactly the Nyquist requency, it is necessary 3 This may sound diicult, but all it means is that you plot the values o a sine wave at discrete intervals o T = :1 seconds. The sampling is done or you by using nt as the time variable in plotting the sine wave, rather than any other arbitrary time-base. Note that whenever you plot a continuous unction with MathCad you are actually plotting its value at a ew discrete points the curve" you see is due to the connection o discrete points by the deault option or displaying a trace. Presentation o discrete or sampled data is made clearer by using other Trace option to explicitly show that the data is discrete, such as bar, point, symbol or stem. 1
11 to introduce a phase shit into the signal in order to see the sampled waveorm. Compare graphs or requencies separated by multiples o the sampling requency i.e. ( + Ns) where N is an integer. Also compare negative and positive requencies in the ormula. At each stage try to justiy what you observe by thinking in the requency domain. 2.2 Discrete-time Systems We are going to investigate the properties o a discrete-time system which has the ollowing inite impulse response: h =:81 h 1 =:247 h 2 =:344 h 3 =:247 h 4 =:81 Using equation 1, plot the output o the system or a unit impulse input i.e. x n = or all values o <n<n (N 1) apart rom one value x m =1,where m =4(or 5 or 6 etc). Veriy that the output o the convolution o x with h is in act the impulse response h given above, but shited in time (remember that Mathcad's arrays run rom zero by deault.) Note that in constructing the convolution ollowing equation 1 an error will be produced i x n k is addressing a non-existent element ox, i.e. n k<. There are a numberoways o tackling this, o which simply starting the convolution at x 4 is the crudest in a real application valuable transient data might be contained in the irst samples o x and by not processing them some inormation may be lost. A little bit o thought and an i(n k ;:::) construction should do the trick (non-existent data rom beore x can be presumed to be zero). Investigate the response o the system to a sampled input sine wave (such as those produced in the irst exercise), changing the requency o the input between zero and the Nyquist requency. Comment on the amplitude and phase delay responses you observe, plotting the amplitude response as a unction o requency. What kind o requency response is this? What unction is h perorming? The extraction o amplitude response can be automated" by using 2D arrays with dimensions o sample number and requency index or both input and output, as or studying orced oscillations and resonance in Worksheet 2. Generate the input data with requencies taken rom an array i with values covering the range 1 Hz. The maximum point on the output waveorm or input requency i can be determined using the max() unction acting on the i th column o output values, which may be extracted using the A <i> operator (see p 156). To avoid transient eects, copy the latter part in time (covering a complete cycle or all requencies) o the output array to a new array beore apply the max() unction. Automation o the phase-delay response is too complex 11
12 to do here, but the approximate behaviour can be determined by visually comparing input and output waveorms or several requencies. Finally, try sampled square waves o dierent requencies, starting with» 5Hz and going up to the Nyquist requency. Comment on what you see, particularly how the shape o the output is dierent to the input (think in terms o the requency components o the square wave input and what eect the transer unction o h will have on each component.) A square wave with period i can be generated using a construction such as: sq n =i(mod(nt; i) <i=2; 1; 1) where the square wave will only be regular i i=2isaninteger multiple o the sampling interval T, so stick to these requencies. Week 6, Session Using the Fast Fourier Transorm Mathcad includes the built-in unctions t() and it() or perorming the Fast Fourier Transorm and Inverse Fast Fourier Transorm respectively (see p 18). Note that t() and it() only work i supplied with a vector o N =2 M samples, and that they then return requency components n = :::N=2. Use the t() unction to analyse a sampled sine wave generated using: x1 n = sin(2ßnt) where n is an integer such that» n» N 1 (N =2 M is the number o samples), is the requency o the input sine wave, and T =1=s (s is the sampling requency and should initially take the same value as N, so that the sampled input covers 1 second o data). Start by using N = 64 and = 6: and comment on what you see. Then try scanning rom 6. to 7. Hz in steps o.25 Hz and comment on what you see. Since the output t() is complex, you should look at the modulus o the individual components. Try changing the number o data points, N, you use without changing the sample requency, s. First ix the input requency to = 6:25 Hz and set the number o samples to 128 and then 256, then ix the requency to =7: Hz and hal the number o samples. Compare how each requency appears to when N = 64 and try to explain any changes you observed. Return to N = 64 and = 7: Hz and try doubling the sample requency, s. Again, comment on what has changed relative to s =64and why. Note that with each new requency it is necessary to avoid the possibility o re-using data in old versions o x rom previous waveorms; this can be done either by renaming the input array or each requency or by explicitly setting its contents to zero beore generating each new sampled sine wave. 12
13 It is particularly important to use a new vector when reducing the number o samples, since once declared, a Mathcad vector can only have its number o elements increased. Think about how the Fourier Transorm is attempting to represent the input in terms o a certain set o requency components. Can you explain why certain values o are ree o spectral leakage? What determines the requency components o the Fast Fourier Transorm? The Fast Fourier Transorm has been presented with samples within a inite time window, but will attempt to analyse the input as i it were an ininte periodic unction which repeats outside the sampling window. Why are some requencies better suited to this treatment than others? I the input waveorm is being multiplied by a rectangular window in the time domain, what is the equivalent picture in the requency domain? It should be clear by now that there are eects which limit the spectral resolution that can be achieved using a FFT. The inite resolution limits our ability to resolve individual spectral components when spectral leakage is present. Look at the FFT spectrum o the sum o two sine waves, given by: z n = sin(2ß 1 nt )+:1 sin(2ß 2 nt ) where 1 = 4:1 and 2 = 7 using N = 64 data points, s = N and T = 1=s. It is very hard to resolve the smaller sine wave because o spectral leakage rom the larger sine wave. So ar, we have eectively been using a rectangular window o width N samples i.e. the sampled sine wave has been multiplied by a unction z n = z n w n, where w n =1 or < n < N 1 and w n = otherwise. Now see what happens when you multiply the data by the Hanning window unction given by: w n =:5+:5 cos( 2ß(n N=2) ): (14) N Plot the windowed version o z n and comment on the dierence between this and the original z n plot (clearest by overlaying them). Then perorm the t() and comment on the changes you see in the requency response and try to explain them, considering the requency domain equivalent o multiplying z n by the windowing unction, w n. You will ind it helpul to compare the requency domain representations o the rectangular and Hanning windows. Week 6, Session Designing a Digital Filter Use equation 13 to calculate the impulse response or a low-pass inite impulse response (FIR) ilter with a normalised cut-o requency w c =2ß=1. Use a rectangular window 13
14 o width 128 to start with. Arrange the centre o the impulse response h n so that it is a maximum at n = 64, and be sure to repair the glitch" in the centre o the sinc unction. Look at the amplitude response o the ilter by using t() to calculate the transer unction corresponding to this impulse response (which has been realised with a inite number o coeicients). Compare this amplitude response with the ideal rom which the impulse response was derived, thinking about the important characteristics o an amplitude response in both the pass band (low requencies), the stop band (high requencies) and the transition between the two. The logarithmic plotting option will allow you to look at the requency response above the cut-o requency. Try changing the width o the rectangular window containing the impulse response (always using 2 M samples), re-centring the impulse response in the new window. Comment on what you see, considering how changes to the rectangular window aect the requency domain. Now return to 128 coeicients and multiply the impulse response by the Hanning window unction Compare the requency response to that o the original 128 coeicient ilter and comment on the changes you see, trying to explain them by thinking in both time and requency domains. 2.5 Using a Digital Filter Finally, add some noise random positive and negative luctuations generated using the rnd() unction to a sine wave. Use your FIR ilter rom the previous exercise to ilter the noisy signal to reduce the noise (in the same way as you iltered various sine waves in exercise 2.2), while preserving the input sine wave (its requency should clearly be below the ilter cut-o requency). Plot the noisy input signal and your iltered output on top o each other and comment on the results. Use the FFT to analyse the requency content o the noisy signal beore and ater it is iltered by the FIR ilter. See i there is any improvement when using the Hanning window unction applied to the impulse response. Suggested reading: It is highly recommended that you do some supplementary reading on digital signal processing beore attempting this worksheet. Search or keywords Digital Signal Processing" or Digital Filters". Some example titles are listed below (any one o these should contain useul analyses o sampling and iltering and there are many similar texts). A. Bateman and W. Yates, Digital Signal Processing Design, Pitman, M. Bellanger, Digital Processing o Signals, 2nd Ed., John Wiley and Sons, J. Dunlop and D.G. Smith, Telecommunications Engineering, 3rd Ed., Chapman and Hall,
15 R.W. Hamming, Digital Filters, 2nd Ed, Prentice-Hall, L.B. Jackson, Digital Filters and Signal Processing, Kluwer, R. Kuc, Introduction to Digital Signal Processing, McGraw-Hill, P.A.Lynn, Introductory Digital Signal Processing with Computer Applications, Wiley, A. Peled, Digital Signal Processing: Theory, Design and Implementation, Wiley,
Complex Spectrum. Box Spectrum. Im f. Im f. Sine Spectrum. Cosine Spectrum 1/2 1/2 1/2. f C -f C 1/2
ECPE 364: view o Small-Carrier Amplitude Modulation his handout is a graphical review o small-carrier amplitude modulation techniques that we studied in class. A Note on Complex Signal Spectra All o the
More informationIntroduction to OFDM. Characteristics of OFDM (Orthogonal Frequency Division Multiplexing)
Introduction to OFDM Characteristics o OFDM (Orthogonal Frequency Division Multiplexing Parallel data transmission with very long symbol duration - Robust under multi-path channels Transormation o a requency-selective
More information3.6 Intersymbol interference. 1 Your site here
3.6 Intersymbol intererence 1 3.6 Intersymbol intererence what is intersymbol intererence and what cause ISI 1. The absolute bandwidth o rectangular multilevel pulses is ininite. The channels bandwidth
More informationExperiment 7: Frequency Modulation and Phase Locked Loops Fall 2009
Experiment 7: Frequency Modulation and Phase Locked Loops Fall 2009 Frequency Modulation Normally, we consider a voltage wave orm with a ixed requency o the orm v(t) = V sin(ω c t + θ), (1) where ω c is
More informationTraditional Analog Modulation Techniques
Chapter 5 Traditional Analog Modulation Techniques Mikael Olosson 2002 2007 Modulation techniques are mainly used to transmit inormation in a given requency band. The reason or that may be that the channel
More informationECE5984 Orthogonal Frequency Division Multiplexing and Related Technologies Fall Mohamed Essam Khedr. Channel Estimation
ECE5984 Orthogonal Frequency Division Multiplexing and Related Technologies Fall 2007 Mohamed Essam Khedr Channel Estimation Matlab Assignment # Thursday 4 October 2007 Develop an OFDM system with the
More informationME scope Application Note 01 The FFT, Leakage, and Windowing
INTRODUCTION ME scope Application Note 01 The FFT, Leakage, and Windowing NOTE: The steps in this Application Note can be duplicated using any Package that includes the VES-3600 Advanced Signal Processing
More informationA Physical Sine-to-Square Converter Noise Model
A Physical Sine-to-Square Converter Noise Model Attila Kinali Max Planck Institute or Inormatics, Saarland Inormatics Campus, Germany adogan@mpi-in.mpg.de Abstract While sinusoid signal sources are used
More informationFourier Signal Analysis
Part 1B Experimental Engineering Integrated Coursework Location: Baker Building South Wing Mechanics Lab Experiment A4 Signal Processing Fourier Signal Analysis Please bring the lab sheet from 1A experiment
More informationOSCILLATORS. Introduction
OSILLATOS Introduction Oscillators are essential components in nearly all branches o electrical engineering. Usually, it is desirable that they be tunable over a speciied requency range, one example being
More informationPLL AND NUMBER OF SAMPLE SYNCHRONISATION TECHNIQUES FOR ELECTRICAL POWER QUALITY MEASURMENTS
XX IMEKO World Congress Metrology or Green Growth September 9 14, 2012, Busan, Republic o Korea PLL AND NUMBER OF SAMPLE SYNCHRONISATION TECHNIQUES FOR ELECTRICAL POWER QUALITY MEASURMENTS Richárd Bátori
More informationEEE 311: Digital Signal Processing I
EEE 311: Digital Signal Processing I Course Teacher: Dr Newaz Md Syur Rahim Associated Proessor, Dept o EEE, BUET, Dhaka 1000 Syllabus: As mentioned in your course calendar Reerence Books: 1 Digital Signal
More informationOverexcitation protection function block description
unction block description Document ID: PRELIMIARY VERSIO ser s manual version inormation Version Date Modiication Compiled by Preliminary 24.11.2009. Preliminary version, without technical inormation Petri
More informationFatigue Life Assessment Using Signal Processing Techniques
Fatigue Lie Assessment Using Signal Processing Techniques S. ABDULLAH 1, M. Z. NUAWI, C. K. E. NIZWAN, A. ZAHARIM, Z. M. NOPIAH Engineering Faculty, Universiti Kebangsaan Malaysia 43600 UKM Bangi, Selangor,
More informationMeasuring the Speed of Light
Physics Teaching Laboratory Measuring the peed o Light Introduction: The goal o this experiment is to measure the speed o light, c. The experiment relies on the technique o heterodyning, a very useul tool
More information1. Motivation. 2. Periodic non-gaussian noise
. Motivation One o the many challenges that we ace in wireline telemetry is how to operate highspeed data transmissions over non-ideal, poorly controlled media. The key to any telemetry system design depends
More informationNoise Removal from ECG Signal and Performance Analysis Using Different Filter
International Journal o Innovative Research in Electronics and Communication (IJIREC) Volume. 1, Issue 2, May 214, PP.32-39 ISSN 2349-442 (Print) & ISSN 2349-45 (Online) www.arcjournal.org Noise Removal
More informationFundamentals of Spectrum Analysis. Christoph Rauscher
Fundamentals o Spectrum nalysis Christoph Rauscher Christoph Rauscher Volker Janssen, Roland Minihold Fundamentals o Spectrum nalysis Rohde & Schwarz GmbH & Co. KG, 21 Mühldorstrasse 15 81671 München Germany
More informationHigh Speed Communication Circuits and Systems Lecture 10 Mixers
High Speed Communication Circuits and Systems Lecture Mixers Michael H. Perrott March 5, 24 Copyright 24 by Michael H. Perrott All rights reserved. Mixer Design or Wireless Systems From Antenna and Bandpass
More informationAnalog ó Digital Conversion Sampled Data Acquisition Systems Discrete Sampling and Nyquist Digital to Analog Conversion Analog to Digital Conversion
Today Analog ó Digital Conversion Sampled Data Acquisition Systems Discrete Sampling and Nyquist Digital to Analog Conversion Analog to Digital Conversion Analog Digital Analog Beneits o digital systems
More informationBiomedical Signals. Signals and Images in Medicine Dr Nabeel Anwar
Biomedical Signals Signals and Images in Medicine Dr Nabeel Anwar Noise Removal: Time Domain Techniques 1. Synchronized Averaging (covered in lecture 1) 2. Moving Average Filters (today s topic) 3. Derivative
More informationSignals. Continuous valued or discrete valued Can the signal take any value or only discrete values?
Signals Continuous time or discrete time Is the signal continuous or sampled in time? Continuous valued or discrete valued Can the signal take any value or only discrete values? Deterministic versus random
More informationSignals and Systems II
1 To appear in IEEE Potentials Signals and Systems II Part III: Analytic signals and QAM data transmission Jerey O. Coleman Naval Research Laboratory, Radar Division This six-part series is a mini-course,
More informationDiscrete Fourier Transform (DFT)
Amplitude Amplitude Discrete Fourier Transform (DFT) DFT transforms the time domain signal samples to the frequency domain components. DFT Signal Spectrum Time Frequency DFT is often used to do frequency
More informationSignal Sampling. Sampling. Sampling. Sampling. Sampling. Sampling
Signal Let s sample the signal at a time interval o Dr. Christopher M. Godrey University o North Carolina at Asheville Photo: C. Godrey Let s sample the signal at a time interval o Reconstruct the curve
More informationarxiv:gr-qc/ v2 13 Jun 2002
LISA data analysis I: Doppler demodulation Neil J. Cornish Department o hysics, Montana State University, Bozeman, MT 59717 Shane L. Larson Space Radiation Laboratory, Caliornia Institute o Technology,
More information2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.
1 2.1 BASIC CONCEPTS 2.1.1 Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2 Time Scaling. Figure 2.4 Time scaling of a signal. 2.1.2 Classification of Signals
More informationProject 2. Project 2: audio equalizer. Fig. 1: Kinter MA-170 stereo amplifier with bass and treble controls.
Introduction Project 2 Project 2: audio equalizer This project aims to motivate our study o ilters by considering the design and implementation o an audio equalizer. An equalizer (EQ) modiies the requency
More informationA new zoom algorithm and its use in frequency estimation
Waves Wavelets Fractals Adv. Anal. 5; :7 Research Article Open Access Manuel D. Ortigueira, António S. Serralheiro, and J. A. Tenreiro Machado A new zoom algorithm and its use in requency estimation DOI.55/wwaa-5-
More informationzt ( ) = Ae find f(t)=re( zt ( )), g(t)= Im( zt ( )), and r(t), and θ ( t) if z(t)=r(t) e
Homework # Fundamentals Review Homework or EECS 562 (As needed or plotting you can use Matlab or another sotware tool or your choice) π. Plot x ( t) = 2cos(2π5 t), x ( t) = 2cos(2π5( t.25)), and x ( t)
More informationSampling and Multirate Techniques for Complex and Bandpass Signals
Sampling and Multirate Techniques or Complex and Bandpass Signals TLT-586/IQ/1 M. Renors, TUT/DCE 21.9.21 Sampling and Multirate Techniques or Complex and Bandpass Signals Markku Renors Department o Communications
More informationLISA data analysis: Doppler demodulation
INSTITUTE OF HYSICSUBLISHING Class. Quantum Grav. 20 (2003) S163 S170 CLASSICAL ANDQUANTUM GRAVITY II: S0264-9381(03)53625-5 LISA data analysis: Doppler demodulation Neil J Cornish 1 and Shane L Larson
More informationAmplifiers. Department of Computer Science and Engineering
Department o Computer Science and Engineering 2--8 Power ampliiers and the use o pulse modulation Switching ampliiers, somewhat incorrectly named digital ampliiers, have been growing in popularity when
More informationChapter 6: Introduction to Digital Communication
93 Chapter 6: Introduction to Digital Communication 6.1 Introduction In the context o this course, digital communications include systems where relatively high-requency analog carriers are modulated y
More informationSimulation of Radio Frequency Integrated Circuits
Simulation o Radio Frequency Integrated Circuits Based on: Computer-Aided Circuit Analysis Tools or RFIC Simulation: Algorithms, Features, and Limitations, IEEE Trans. CAS-II, April 2000. Outline Introduction
More informationSolid State Relays & Its
Solid State Relays & Its Applications Presented By Dr. Mostaa Abdel-Geliel Course Objectives Know new techniques in relay industries. Understand the types o static relays and its components. Understand
More informationWith the proposed technique, those two problems will be overcome. reduction is to eliminate the specific harmonics, which are the lowest orders.
CHAPTER 3 OPTIMIZED HARMONIC TEPPED-WAVEFORM TECHNIQUE (OHW The obective o the proposed optimized harmonic stepped-waveorm technique is to reduce, as much as possible, the harmonic distortion in the load
More informationThe Research of Electric Energy Measurement Algorithm Based on S-Transform
International Conerence on Energy, Power and Electrical Engineering (EPEE 16 The Research o Electric Energy Measurement Algorithm Based on S-Transorm Xiyang Ou1,*, Bei He, Xiang Du1, Jin Zhang1, Ling Feng1,
More informationThe Discrete Fourier Transform. Claudia Feregrino-Uribe, Alicia Morales-Reyes Original material: Dr. René Cumplido
The Discrete Fourier Transform Claudia Feregrino-Uribe, Alicia Morales-Reyes Original material: Dr. René Cumplido CCC-INAOE Autumn 2015 The Discrete Fourier Transform Fourier analysis is a family of mathematical
More informationSoftware Defined Radio Forum Contribution
Committee: Technical Sotware Deined Radio Forum Contribution Title: VITA-49 Drat Speciication Appendices Source Lee Pucker SDR Forum 604-828-9846 Lee.Pucker@sdrorum.org Date: 7 March 2007 Distribution:
More informationDepartment of Electronic Engineering NED University of Engineering & Technology. LABORATORY WORKBOOK For the Course SIGNALS & SYSTEMS (TC-202)
Department of Electronic Engineering NED University of Engineering & Technology LABORATORY WORKBOOK For the Course SIGNALS & SYSTEMS (TC-202) Instructor Name: Student Name: Roll Number: Semester: Batch:
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationA MATLAB Model of Hybrid Active Filter Based on SVPWM Technique
International Journal o Electrical Engineering. ISSN 0974-2158 olume 5, Number 5 (2012), pp. 557-569 International Research Publication House http://www.irphouse.com A MATLAB Model o Hybrid Active Filter
More informationISSUE: April Fig. 1. Simplified block diagram of power supply voltage loop.
ISSUE: April 200 Why Struggle with Loop ompensation? by Michael O Loughlin, Texas Instruments, Dallas, TX In the power supply design industry, engineers sometimes have trouble compensating the control
More informationInstantaneous frequency Up to now, we have defined the frequency as the speed of rotation of a phasor (constant frequency phasor) φ( t) = A exp
Exponential modulation Instantaneous requency Up to now, we have deined the requency as the speed o rotation o a phasor (constant requency phasor) φ( t) = A exp j( ω t + θ ). We are going to generalize
More informationOutline. Wireless PHY: Modulation and Demodulation. Admin. Page 1. g(t)e j2πk t dt. G[k] = 1 T. G[k] = = k L. ) = g L (t)e j2π f k t dt.
Outline Wireless PHY: Modulation and Demodulation Y. Richard Yang Admin and recap Basic concepts o modulation Amplitude demodulation requency shiting 09/6/202 2 Admin First assignment to be posted by this
More informationFFT analysis in practice
FFT analysis in practice Perception & Multimedia Computing Lecture 13 Rebecca Fiebrink Lecturer, Department of Computing Goldsmiths, University of London 1 Last Week Review of complex numbers: rectangular
More informationECEN 5014, Spring 2013 Special Topics: Active Microwave Circuits and MMICs Zoya Popovic, University of Colorado, Boulder
ECEN 5014, Spring 2013 Special Topics: Active Microwave Circuits and MMICs Zoya Popovic, University o Colorado, Boulder LECTURE 13 PHASE NOISE L13.1. INTRODUCTION The requency stability o an oscillator
More information(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters
FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according
More informationDARK CURRENT ELIMINATION IN CHARGED COUPLE DEVICES
DARK CURRENT ELIMINATION IN CHARGED COUPLE DEVICES L. Kňazovická, J. Švihlík Department o Computing and Control Engineering, ICT Prague Abstract Charged Couple Devices can be ound all around us. They are
More informationSpectrum Analysis - Elektronikpraktikum
Spectrum Analysis Introduction Why measure a spectra? In electrical engineering we are most often interested how a signal develops over time. For this time-domain measurement we use the Oscilloscope. Like
More informationExploring QAM using LabView Simulation *
OpenStax-CNX module: m14499 1 Exploring QAM using LabView Simulation * Robert Kubichek This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 1 Exploring
More informationData Acquisition Systems. Signal DAQ System The Answer?
Outline Analysis of Waveforms and Transforms How many Samples to Take Aliasing Negative Spectrum Frequency Resolution Synchronizing Sampling Non-repetitive Waveforms Picket Fencing A Sampled Data System
More informationDFT: Discrete Fourier Transform & Linear Signal Processing
DFT: Discrete Fourier Transform & Linear Signal Processing 2 nd Year Electronics Lab IMPERIAL COLLEGE LONDON Table of Contents Equipment... 2 Aims... 2 Objectives... 2 Recommended Textbooks... 3 Recommended
More informationI am very pleased to teach this class again, after last year s course on electronics over the Summer Term. Based on the SOLE survey result, it is clear that the format, style and method I used worked with
More informationIslamic University of Gaza. Faculty of Engineering Electrical Engineering Department Spring-2011
Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#4 Sampling and Quantization OBJECTIVES: When you have completed this assignment,
More informationSEG/San Antonio 2007 Annual Meeting. Summary. Morlet wavelet transform
Xiaogui Miao*, CGGVeritas, Calgary, Canada, Xiao-gui_miao@cggveritas.com Dragana Todorovic-Marinic and Tyler Klatt, Encana, Calgary Canada Summary Most geologic changes have a seismic response but sometimes
More informationUnderstanding Digital Signal Processing
Understanding Digital Signal Processing Richard G. Lyons PRENTICE HALL PTR PRENTICE HALL Professional Technical Reference Upper Saddle River, New Jersey 07458 www.photr,com Contents Preface xi 1 DISCRETE
More information+ a(t) exp( 2πif t)dt (1.1) In order to go back to the independent variable t, we define the inverse transform as: + A(f) exp(2πif t)df (1.
Chapter Fourier analysis In this chapter we review some basic results from signal analysis and processing. We shall not go into detail and assume the reader has some basic background in signal analysis
More informationThe fourier spectrum analysis of optical feedback self-mixing signal under weak and moderate feedback
University o Wollongong Research Online Faculty o Inormatics - Papers (Archive) Faculty o Engineering and Inormation Sciences 8 The ourier spectrum analysis o optical eedback sel-mixing signal under weak
More informationTheory of Telecommunications Networks
Theory of Telecommunications Networks Anton Čižmár Ján Papaj Department of electronics and multimedia telecommunications CONTENTS Preface... 5 1 Introduction... 6 1.1 Mathematical models for communication
More informationFurther developments on gear transmission monitoring
Further developments on gear transmission monitoring Niola V., Quaremba G., Avagliano V. Department o Mechanical Engineering or Energetics University o Naples Federico II Via Claudio 21, 80125, Napoli,
More informationChapter 5 Window Functions. periodic with a period of N (number of samples). This is observed in table (3.1).
Chapter 5 Window Functions 5.1 Introduction As discussed in section (3.7.5), the DTFS assumes that the input waveform is periodic with a period of N (number of samples). This is observed in table (3.1).
More informationSystem analysis and signal processing
System analysis and signal processing with emphasis on the use of MATLAB PHILIP DENBIGH University of Sussex ADDISON-WESLEY Harlow, England Reading, Massachusetts Menlow Park, California New York Don Mills,
More informationOutline. Wireless PHY: Modulation and Demodulation. Admin. Page 1. G[k] = 1 T. g(t)e j2πk t dt. G[k] = = k L. ) = g L (t)e j2π f k t dt.
Outline Wireless PHY: Modulation and Demodulation Y. Richard Yang Admin and recap Basic concepts o modulation Amplitude modulation Amplitude demodulation requency shiting 9/6/22 2 Admin First assignment
More informationELEC3106 Electronics. Lecture notes: non-linearity and noise. Objective. Non-linearity. Non-linearity measures
ELEC316 Electronics Lecture notes: non-linearity and noise Objective The objective o these brie notes is to supplement the textbooks used in the course on the topic o non-linearity and electrical noise.
More informationThe Fast Fourier Transform
The Fast Fourier Transform Basic FFT Stuff That s s Good to Know Dave Typinski, Radio Jove Meeting, July 2, 2014, NRAO Green Bank Ever wonder how an SDR-14 or Dongle produces the spectra that it does?
More informationLecture 2: SIGNALS. 1 st semester By: Elham Sunbu
Lecture 2: SIGNALS 1 st semester 1439-2017 1 By: Elham Sunbu OUTLINE Signals and the classification of signals Sine wave Time and frequency domains Composite signals Signal bandwidth Digital signal Signal
More informationstate the transfer function of the op-amp show that, in the ideal op-amp, the two inputs will be equal if the output is to be finite
NTODUCTON The operational ampliier (op-amp) orms the basic building block o many analogue systems. t comes in a neat integrated circuit package and is cheap and easy to use. The op-amp gets its name rom
More informationNew metallic mesh designing with high electromagnetic shielding
MATEC Web o Conerences 189, 01003 (018) MEAMT 018 https://doi.org/10.1051/mateccon/01818901003 New metallic mesh designing with high electromagnetic shielding Longjia Qiu 1,,*, Li Li 1,, Zhieng Pan 1,,
More informationBasic Signals and Systems
Chapter 2 Basic Signals and Systems A large part of this chapter is taken from: C.S. Burrus, J.H. McClellan, A.V. Oppenheim, T.W. Parks, R.W. Schafer, and H. W. Schüssler: Computer-based exercises for
More informationAN EFFICIENT SET OF FEATURES FOR PULSE REPETITION INTERVAL MODULATION RECOGNITION
AN EFFICIENT SET OF FEATURES FOR PULSE REPETITION INTERVAL MODULATION RECOGNITION J-P. Kauppi, K.S. Martikainen Patria Aviation Oy, Naulakatu 3, 33100 Tampere, Finland, ax +358204692696 jukka-pekka.kauppi@patria.i,
More informationComplex RF Mixers, Zero-IF Architecture, and Advanced Algorithms: The Black Magic in Next-Generation SDR Transceivers
Complex RF Mixers, Zero-F Architecture, and Advanced Algorithms: The Black Magic in Next-Generation SDR Transceivers By Frank Kearney and Dave Frizelle Share on ntroduction There is an interesting interaction
More information6.02 Practice Problems: Modulation & Demodulation
1 of 12 6.02 Practice Problems: Modulation & Demodulation Problem 1. Here's our "standard" modulation-demodulation system diagram: at the transmitter, signal x[n] is modulated by signal mod[n] and the
More informationSampling and Signal Processing
Sampling and Signal Processing Sampling Methods Sampling is most commonly done with two devices, the sample-and-hold (S/H) and the analog-to-digital-converter (ADC) The S/H acquires a continuous-time signal
More informationECE 5655/4655 Laboratory Problems
Assignment #4 ECE 5655/4655 Laboratory Problems Make Note o the Following: Due Monday April 15, 2019 I possible write your lab report in Jupyter notebook I you choose to use the spectrum/network analyzer
More informationDigital Video and Audio Processing. Winter term 2002/ 2003 Computer-based exercises
Digital Video and Audio Processing Winter term 2002/ 2003 Computer-based exercises Rudolf Mester Institut für Angewandte Physik Johann Wolfgang Goethe-Universität Frankfurt am Main 6th November 2002 Chapter
More informationSystem on a Chip. Prof. Dr. Michael Kraft
System on a Chip Prof. Dr. Michael Kraft Lecture 5: Data Conversion ADC Background/Theory Examples Background Physical systems are typically analogue To apply digital signal processing, the analogue signal
More informationFundamentals of Time- and Frequency-Domain Analysis of Signal-Averaged Electrocardiograms R. Martin Arthur, PhD
CORONARY ARTERY DISEASE, 2(1):13-17, 1991 1 Fundamentals of Time- and Frequency-Domain Analysis of Signal-Averaged Electrocardiograms R. Martin Arthur, PhD Keywords digital filters, Fourier transform,
More informationHarmonic Analysis. Purpose of Time Series Analysis. What Does Each Harmonic Mean? Part 3: Time Series I
Part 3: Time Series I Harmonic Analysis Spectrum Analysis Autocorrelation Function Degree of Freedom Data Window (Figure from Panofsky and Brier 1968) Significance Tests Harmonic Analysis Harmonic analysis
More informationMotor Gear Fault Diagnosis by Current, Noise and Vibration on AC Machine Considering Environment Sun-Ki Hong, Ki-Seok Kim, Yong-Ho Cho
Motor Gear Fault Diagnosis by Current, Noise and Vibration on AC Machine Considering Environment Sun-Ki Hong, Ki-Seok Kim, Yong-Ho Cho Abstract Lots o motors have been being used in industry. Thereore
More informationThe Fundamentals of Mixed Signal Testing
The Fundamentals of Mixed Signal Testing Course Information The Fundamentals of Mixed Signal Testing course is designed to provide the foundation of knowledge that is required for testing modern mixed
More informationCOMP 558 lecture 5 Sept. 22, 2010
Up to now, we have taken the projection plane to be in ront o the center o projection. O course, the physical projection planes that are ound in cameras (and eyes) are behind the center o the projection.
More informationEE 791 EEG-5 Measures of EEG Dynamic Properties
EE 791 EEG-5 Measures of EEG Dynamic Properties Computer analysis of EEG EEG scientists must be especially wary of mathematics in search of applications after all the number of ways to transform data is
More informationA Detailed Lesson on Operational Amplifiers - Negative Feedback
07 SEE Mid tlantic Section Spring Conerence: Morgan State University, Baltimore, Maryland pr 7 Paper ID #0849 Detailed Lesson on Operational mpliiers - Negative Feedback Dr. Nashwa Nabil Elaraby, Pennsylvania
More informationQäf) Newnes f-s^j^s. Digital Signal Processing. A Practical Guide for Engineers and Scientists. by Steven W. Smith
Digital Signal Processing A Practical Guide for Engineers and Scientists by Steven W. Smith Qäf) Newnes f-s^j^s / *" ^"P"'" of Elsevier Amsterdam Boston Heidelberg London New York Oxford Paris San Diego
More informationEE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM
EE 215 Semester Project SPECTRAL ANALYSIS USING FOURIER TRANSFORM Department of Electrical and Computer Engineering Missouri University of Science and Technology Page 1 Table of Contents Introduction...Page
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 1 FIR as a
More information2) How fast can we implement these in a system
Filtration Now that we have looked at the concept of interpolation we have seen practically that a "digital filter" (hold, or interpolate) can affect the frequency response of the overall system. We need
More informationDSP APPLICATION TO THE PORTABLE VIBRATION EXCITER
DSP PPLICTION TO THE PORTBLE VIBRTION EXCITER W. Barwicz 1, P. Panas 1 and. Podgórski 2 1 Svantek Ltd., 01-410 Warsaw, Poland Institute o Radioelectronics, Faculty o Electronics and Inormation Technology
More informationTIME-FREQUENCY ANALYSIS OF NON-STATIONARY THREE PHASE SIGNALS. Z. Leonowicz T. Lobos
Copyright IFAC 15th Triennial World Congress, Barcelona, Spain TIME-FREQUENCY ANALYSIS OF NON-STATIONARY THREE PHASE SIGNALS Z. Leonowicz T. Lobos Wroclaw University o Technology Pl. Grunwaldzki 13, 537
More informationB.Tech III Year II Semester (R13) Regular & Supplementary Examinations May/June 2017 DIGITAL SIGNAL PROCESSING (Common to ECE and EIE)
Code: 13A04602 R13 B.Tech III Year II Semester (R13) Regular & Supplementary Examinations May/June 2017 (Common to ECE and EIE) PART A (Compulsory Question) 1 Answer the following: (10 X 02 = 20 Marks)
More informationFunctional Description of Algorithms Used in Digital Receivers
> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < 1 Functional Description o Algorithms Used in Digital Receivers John Musson #, Old Dominion University, Norolk, VA
More informationTRANSFORMS / WAVELETS
RANSFORMS / WAVELES ransform Analysis Signal processing using a transform analysis for calculations is a technique used to simplify or accelerate problem solution. For example, instead of dividing two
More informationSAMPLING THEORY. Representing continuous signals with discrete numbers
SAMPLING THEORY Representing continuous signals with discrete numbers Roger B. Dannenberg Professor of Computer Science, Art, and Music Carnegie Mellon University ICM Week 3 Copyright 2002-2013 by Roger
More informationLecture Schedule: Week Date Lecture Title
http://elec3004.org Sampling & More 2014 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 1 2-Mar Introduction 3-Mar
More informationObjectives. Presentation Outline. Digital Modulation Lecture 03
Digital Modulation Lecture 03 Inter-Symbol Interference Power Spectral Density Richard Harris Objectives To be able to discuss Inter-Symbol Interference (ISI), its causes and possible remedies. To be able
More informationSpread-Spectrum Technique in Sigma-Delta Modulators
Spread-Spectrum Technique in Sigma-Delta Modulators by Eric C. Moule Submitted in Partial Fulillment o the Requirements or the Degree Doctor o Philosophy Supervised by Proessor Zeljko Ignjatovic Department
More information6 Sampling. Sampling. The principles of sampling, especially the benefits of coherent sampling
Note: Printed Manuals 6 are not in Color Objectives This chapter explains the following: The principles of sampling, especially the benefits of coherent sampling How to apply sampling principles in a test
More informationSIGNALS AND SYSTEMS LABORATORY 13: Digital Communication
SIGNALS AND SYSTEMS LABORATORY 13: Digital Communication INTRODUCTION Digital Communication refers to the transmission of binary, or digital, information over analog channels. In this laboratory you will
More information