Unreferenced date December 2009 The Whittaker Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous time bandlimited signal from a set of equally spaced samples. Definition The interpolation formula , as it is commonly called, dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called Shannon s interpolation formula and Whittaker s interpolation formula . E. T. Whittaker, who published it in 1915, called it the Cardinal series . The sampling theorem states that, under certain limiting conditions , a function x t can be recovered exactly from its samples, x n x nT , by the Whittaker Shannon interpolation formula math x t sum n infty infty x n cdot rm sinc left frac t nT T right , math where T 1 f sub s sub is the sampling interval , f sub s sub is the sampling rate , and sinc x is the normalized sinc function . Validity condition Image bandlimited.svg thumb right 240px Spectrum of a bandlimited signal as a function of frequency. The two sided bandwidth R sub N sub 2 B is known as the Nyquist rate for the signal. If the function x t is bandlimited, and sampled at a high enough rate, the interpolation formula is guaranteed to reconstruct it exactly. Formally, if there exists some B 0 such that the function x t is bandlimited to bandwidth B that is, it has a Fourier transform math scriptstyle mathcal F x t X f 0 math for f B and the sampling rate , f sub s sub , exceeds the Nyquist rate , twice the bandwidth f sub s sub 2 B . Equivalently math T frac 1 2B math then the interpolation formula will exactly reconstruct the original x t from its samples. Otherwise, aliasing may occur that is, frequencies at or above f sub s sub 2 may be erroneously reconstructed. See Aliasing for further discussion on this point. Interpolation as convolution sum The ... more details
DSF can mean Deutsches Sportfernsehen , a German TV channel for sport Democratic Students Front, an independent political students organisation of Jadavpur University DSF Refractories & Minerals Ltd , a British refractory brick manufacturer David Suzuki Foundation , a Canadian environmental charity Delaware State Fair Driver San Francisco , an upcoming video game Digital solidarity fund , a Swiss law foundation which aims to reduce the digital divide Director Special Forces , the British Major General commanding United Kingdom Special Forces Dubai Shopping Festival , a prime festival retail season in Dubai Discrete Summation Formulae , a number of formulas for synthesizing bandlimited periodic signal s Double skin facade , a type of building facade Delivery Sequence File , a database which contains every mail address serviced by the USPS Douglas S. Freeman High School , a high school in Richmond, Virginia Django Software Foundation , a non profit foundation to support Django web framework Django a web programming framework development Device Simulation Framework, a part of the Windows Driver Kit disambig bs DSF da DSF de DSF fr DSF it DSF nl DSF sv DSF ... more details
distinguish Nyquist frequency Image Bandlimited.svg thumb Spectrum of a bandlimited signal as a function of frequency right 240px In signal processing , the Nyquist rate , named after Harry Nyquist , is two times the Bandwidth signal processing bandwidth of a bandlimited signal or a bandlimited channel. This term is used to mean two different things under two different circumstances as a lower bound for the sample rate for alias free signal sampling ref cite book title Design of multi bit delta sigma A D converters author Yves Geerts, Michiel Steyaert, and Willy Sansen publisher Springer year 2002 ISBN 1402070780 url http books.google.com books?id 99nwX ovq7QC&pg PA15&dq nyquist rate define bandwidth sampling PPA14,M1 ref not to be confused with the Nyquist frequency , which is half the sampling rate of a discrete time system and as an upper bound for the symbol rate across a bandwidth limited baseband channel such as a telegraph line ref cite book title Telecommunication System Engineering author Roger L. Freeman publisher John Wiley & Sons year 2004 ISBN 0471451339 url http books.google.com books?id Ga7PYE7E8kQC&pg PA399&dq nyquist rate define bandwidth symbols pages 399 ref or passband channel such as a limited radio frequency band or a frequency division multiplex channel. Nyquist rate relative to sampling The Nyquist rate is the minimum sampling frequency sampling rate required to avoid aliasing , equal to twice the highest frequency contained within the signal. math f N stackrel mathrm def 2 B , math where math B , math is the highest frequency at which the signal can have nonzero energy. To avoid aliasing, the sampling rate must exceed the Nyquist rate math f S f N. , math A signal whose positive frequency range of significant energy is 0, B , as depicted above, is called Baseband baseband or lowpass . But when the frequency range is A , B , for some A B &minus A , it is called bandpass . In that case, aliasing is not necessarily ... more details
the signal being sampled also has periodic content. Bandlimited functions main Nyquist Shannon sampling .... Functions whose frequency content is bounded bandlimited have infinite duration. If sampled ... two at 1760 Hz A6 . The sawtooths alternate between bandlimited non aliased sawtooths and aliased sawtooths and the sampling rate is 22.05 kHz. The bandlimited sawtooths are synthesized ... fundamental frequencies, and while the bandlimited sawtooth is still clear at 1760 Hz, the aliased .... Listen filename Sawtooth aliasingdemo.ogg title Sawtooth aliasing demo description 440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased format ... more details
domain . Practically, this means that a signal must be bandlimited to less than half the sampling ... performed on discrete time signals must be bandlimited to F sub s sub 2 epsilon . Wagner s book ... more details
Image Bandlimited.svg thumb Fig.1 Hypothetical spectrum of a bandlimited signal as a function of frequency ... that a bandlimited analog signal that has been sampled can be perfectly reconstructed from an infinite ... time any time limited x t cannot be perfectly BandlimitedBandlimited versus timelimited bandlimited . Perfect reconstruction is mathematically possible for the idealized model but only an approximation ... yields a sub Nyquist sampling criterion. Introduction A signal or function is bandlimited if it contains ... signal processing bandwidth B . The sampling theorem asserts that, given such a bandlimited signal ... mathrm def int infty infty x t e i 2 pi f t rm d t. math The signal x t is said to be bandlimited ... rate and is a property of the bandlimited signal, while f sub s sub 2 is called the Nyquist ... method that in theory exactly reconstructs any given bandlimited x t with any bandlimit ... , denoted f sub N sub . Both of these cases imply that the signal to be sampled must be bandlimited ... signal is bandlimited such that frequency components at or above half of the sampling frequency ... never be perfectly bandlimited, since ideal brick wall filters cannot be realized. All practical filters ... limited signal can never be bandlimited. This means that even if an ideal reconstruction ... thumb Fig.3 Hypothetical spectrum of a properly sampled bandlimited signal blue and images ... thumb Fig.4 Top Hypothetical spectrum of an insufficiently sampled bandlimited signal blue ... a spectrum unlike the original. Bottom Hypothetical spectrum of a marginally sufficiently sampled bandlimited ... device is not in general bandlimited, and the non ideal sampling is itself a useful type of low ... f , of a properly sampled bandlimited signal blue and images green that do not overlap. A brick wall ... not say whether the sampling theorem extends to bandlimited stationary random processes. Sampling ... in two parts, as done above the first stating the fact that a bandlimited function is completely determined ... more details
with the received signal. Bandlimited channels Another cause of intersymbol interference is the transmission of a signal through a bandlimited channel, i.e., one where the frequency response ... with following symbols. As opposed to multipath propagation, bandlimited channels are present ... data over bandlimited channels usually implement pulse shaping to avoid interference caused by the bandwidth ... more details
Image sawtooth td and fd.png thumb 400px A bandlimited sawtooth wave pictured in the time domain top and frequency domain bottom . The fundamental is at 220 Hz A3 . The sawtooth wave or saw wave is a kind of non sinusoidal waveform . It is named a sawtooth based on its resemblance to the teeth on the blade of a saw . The convention is that a sawtooth wave ramps upward and then sharply drops. However, there are also sawtooth waves in which the wave ramps downward and then sharply rises. The latter type of sawtooth wave is called a reverse sawtooth wave or inverse sawtooth wave . The piecewise linear function math x t t lfloor t rfloor t operatorname floor t math based on the floor function of time t is an example of a sawtooth wave with Wave period period 1. A more general form, in the range 1 to 1, and with period a , is math x t 2 left t over a left lfloor t over a 1 over 2 right rfloor right math math 2 left t over a operatorname floor left t over a 1 over 2 right right math This sawtooth function has the same Phase waves phase as the sine function. A sawtooth wave s sound is harsh and clear and its spectrum contains both even and odd harmonic s of the fundamental frequency . Because it contains all the integer harmonics, it is one of the best waveforms to use for synthesizing musical sounds, particularly bowed string instruments like violins and cellos, using subtractive synthesis . A sawtooth can be constructed using additive synthesis . The infinite Fourier series math x mathrm sawtooth t frac 2 pi sum k 1 infin 1 k frac sin 2 pi kft k math converges to an inverse sawtooth wave. A conventional sawtooth can be constructed using math x mathrm sawtooth t frac 2 pi sum k ... . If the waveform is digitally created directly in the time domain using a non bandlimited form, such as y ... at 440 Hz A4 and 880 Hz A5 and 1760 Hz A6 is available below. Both bandlimited non aliased ... demo description Sawtooth waves played bandlimited and aliased at 440 Hz, 880 Hz, and 1760 ... more details
unreferenced date September 2010 The problem of reconstruction from zero crossings can be stated as given the zero crossing s of a Continuous signal continuous signal information theory signal , is it possible to reconstruct the signal to within a constant factor ? Worded differently, what are the conditions under which a signal can be reconstructed from its zero crossings? This problem has 2 parts. Firstly proving that there is a unique reconstruction of the signal from the zero crossings and secondly how to actually go about reconstructing the signal. Though there have been quite a few attempts before, no conclusive solution has been found. Ben Logan from the Bell laboratories wrote a paper in 1977 in the Bell Systems Technical Journal giving some criteria under which unique reconstruction is possible. Though this has been a major step towards the solution, many people are dissatisfied with the type of condition which results from his paper. According to Logan a signal is uniquely reconstructible from its zero crossings if The signal x t and its Hilbert transform x sup t sup have no zeros in common with each other. The frequency domain representation of the signal is at most 1 octave long, in other words, it is bandpass bandlimited limited between some B and 2B. Further reading BF Logan, Jr. Information in the Zero Crossings of Bandpass Signals , Bell System Technical. Journal , vol. 56, pp. 487 510, April 1977 External links http www.rle.mit.edu dspg documents recon2dim 1985.pdf Reconstruction of two dimensional signals from threshold crossings Category Signal processing ... more details
A triangle wave is a non sinusoidal waveform named for its Triangle triangular shape. Image triangle td and fd.png thumb 400px A bandlimited triangle wave pictured in the time domain top and frequency domain bottom . The fundamental is at 220 Hz A3 . Like a square wave , the triangle wave contains only odd harmonic s. However, the higher harmonics roll off roll off much faster than in a square wave proportional to the inverse square of the harmonic number as opposed to just the inverse . It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every 4n 1 th harmonic by 1 or changing its phase by , and rolling off the harmonics by the inverse square of their relative frequency to the Fundamental frequency fundamental . This infinite Fourier series converges to the triangle wave math begin align x mathrm triangle t & frac 8 pi 2 sum k 0 infty 1 k , frac sin left 2k 1 omega t right 2k 1 2 & frac 8 pi 2 left sin omega t 1 over 9 sin 3 omega t 1 over 25 sin 5 omega t cdots right end align math where math scriptstyle omega math is the angular frequency . Image Synthesis triangle.gif thumb 350px right Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See Fourier Transform Fourier Analysis for a mathematical description. Listen filename 220 Hz anti aliased triangle wave.ogg title Triangle wave sound sample description 5 seconds of anti aliasing anti aliased triangle wave at 220 Hz format Ogg Another definition of the triangle wave, with range from 1 to 1 and period 2 a is math x t frac 2 a left t a left lfloor frac t a frac 1 2 right rfloor right 1 left lfloor frac t a frac 1 2 right rfloor math where the symbol math scriptstyle lfloor n rfloor math represent the Floor and ceiling functions floor function of n . Also, the triangle wave can be the absolute value of the sawtooth wave br math x t left 2 left t over a left lfloor t over a 1 over 2 right rfloor rig ... more details
In a mixed signal system analog signal analog and digital signal digital , a reconstruction filter or anti imaging filter is used to construct a smooth analogue signal from a digital input, as in the case of a digital to analogue converter Digital to analog converter DAC or other sampled data output device. Sampled data reconstruction filters The sampling theorem describes why the input of an Analog to digital converter ADC requires a low pass analog electronic filter , called the anti aliasing filter the sampled input signal must be bandlimited to prevent aliasing here meaning waves of higher frequency being recorded as a lower frequency . For the same reason, the output of a DAC requires a low pass analog filter, called a reconstruction filter, as the output signal must be bandlimited, to prevent aliasing here meaning Fourier coefficients being reconstructed as low frequency waves, not as higher frequency aliases , as in the Whittaker Shannon interpolation formula . Ideally, both filters should be brickwall filter s, constant phase delay in the pass band with constant flat frequency response, and zero response from the Nyquist frequency . This is given by a filter with a Sinc function sinc impulse response. Implementation Practical filters have non flat frequency or phase response in the pass band and incomplete suppression of the signal elsewhere, as a sinc waveform has an infinite response to a signal, in both the positive and negative time directions, which is impossible to perform in real time it would require infinite delay. In systems that have both, the anti aliasing filter and a reconstruction filter may be of identical design. For example, both the input and the output for audio equipment is sampled at 44.1 kHz. Both audio filter s block as much as possible above 22 kHz and pass as much as possible below 20 kHz. Typically both filters are active op amp filters, with exactly the same selection of resistors and capacitors. Whilst in theory a DAC gives a ser ... more details
be bandlimited. However, in practice, no time limited signal can be bandlimited. Since ... device in question , it follows that they are not bandlimited. However, by designing a sampler ... more details
of bandlimited functions , Proceedings of the International Conference on Sampling Theory and Applications ..., Vol. 2 1995 ,pp. 160 172. Sampling theorem for signals bandlimited to a general domain in several ... theorem for functions bandlimited to a disc, The journal of Complex Variables Theory and Applications ... more details
systems, bandlimited AWGN cannot be ignored. When modeling bandlimited AWGN in the phasor domain ... distributed from 0 to 2 . The graph to the right shows an example of how bandlimited AWGN can ... more details
File High accuracy settling time measurements figure 1.png thumb An illustration of overshoot, followed by ringing signal ringing and settle time . In signal processing , control theory , electronics , and mathematics , overshoot is when a signal or function exceeds its target. It arises especially in the step response of bandlimited systems such as low pass filter s. It is often followed by ringing signal ringing , and at times conflated with this latter. Definition Maximum Overshoot Overshoot signal is defined in Katsuhiko Ogata s Discrete time control systems as the maximum peak value of the response curve measured from the desired response of the system. ref Cite book title Discrete time control systems first Katsuhiko last Ogata publisher Prentice Hall year 1987 isbn 0132161028 page 344 ref Control theory In control theory , overshoot refers to an output exceeding its final, steady state value. ref name Kuo cite book author Kuo, Benjamin C & Golnaraghi M F title Automatic control systems edition Eighth edition page 7.3 p. 236 237 year 2003 publisher Wiley location NY isbn 0471134767 url http worldcat.org isbn 0471134767 ref For a step response step input , the percentage overshoot PO is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. Also see the definition of overshoot in an Electronics electronics context . I ll try to remember to generate an image when I m at work tomorrow, but I might not get back to this for days. Feel free to do as you see fit The percentage overshoot is a function of the Damping ratio and is given by math PO 100 cdot e left frac zeta pi sqrt 1 zeta 2 right math The damping ratio can also be found by math zeta sqrt frac ln PO 2 pi 2 ln PO 2 math Electronics Image clock signal.gif 300 px thumb Overshoot and undershoot in electronics electronic Signal electrical engineering signal . In electronics, overshoot refers to th ... more details
14 . Amplitude modulated signals are modeled as the product of a bandlimited message waveform, u sub ... filter.tif thumb 400px right Filter whose frequency response is bandlimited to about 95 of the Nyquist ... bandlimited transfer function math sigma H omega begin cases e i pi 2 , & pi omega 0 e i pi 2 , & 0 ... cases math If a signal math scriptstyle u t math is bandlimited , then math scriptstyle H u t math is bandlimited in the same way. Consequently, both these signals can be sampled according to the sampling ... more details
Encyclopedia
top
