it fourier transforms as used in signal theory

... done with 2D − multiplications But can it be done with fewer multiplications? Whereas complexity theory has huge gaps in almost all areas, it has triumphed in convolution algebras The minimum ... previous case So far intuition proved accurate In complexity theory one learns early on to be skeptical of intuitions An early surprising result in complexity theory — and to date still one of its ... obtained In the past several years new, more powerful, processors have been introduced Sun and HP have incorporated new vector instructions Intel has introduced its aggressive Intel’s MMX architecture...

Ngày tải lên: 25/10/2013, 02:15

... techniques for carrying out Fourier transforms of suitable functions without using integration In the rest of the book the definitions and results given here will be used to obtain useful results ... function, with peak value at y = (This convolution is shown in Figure 3.4, with A = and T = in this case.) 28 Fourier Transforms in Radar and Signal Processing ∞ ͐−∞ sinc (x − m ) sinc (x − n ... y ), and this is Rule 6b However, in this case, the result is easily obtained from the definitions of the Fourier transform in (2.2), as shown in Appendix 2B In Table 2.2, not only are pairs 1b,...

Ngày tải lên: 18/06/2014, 10:05

... overlapping sinc functions These are shown as dotted lines in Figure 3.9(b), with the pulse spectrum as the solid line The 48 Fourier Transforms in Radar and Signal Processing Figure 3.9 Raised cosine ... an interesting exercise in the use of the rules-and-pairs method, showing that the method gives a solution for the spectrum quite 39 40 Fourier Transforms in Radar and Signal Processing easily ... the poor spectrum, with high side lobes This discontinuity in level is removed by generating rising and falling edges of finite slope In the case of the symmetric trapezoidal pulse, this is achieved...

Ngày tải lên: 18/06/2014, 10:05

... the information in the waveform is retained by sampling it at a rate 2W (or higher) In principle, reconstituting the waveform in this case is achieved by driving a rectangular bandwidth low-pass ... (3.30) 58 Fourier Transforms in Radar and Signal Processing As a check, we note that if we used a single rounding function r , with transform R , the expression in (3.30) reduces to R( f ) ͩ sinc f ... (k integral, ≤ ␣ < 1), then all the information in the waveform is retained by sampling it at a rate 2rW, where r is given in (4.13) above 74 Fourier Transforms in Radar and Signal Processing...

Ngày tải lên: 18/06/2014, 10:05

... that it must remain finite), as there is no spectral power in these regions Thus we are free to choose G to be of any form as long as it satisfies the conditions in (5.7) In the case of sampling ... Hilbert sampling and quadrature sampling, discussed 82 Fourier Transforms in Radar and Signal Processing above, in particular from the point of view of finding the minimum sampling rate needed ... retain the signal information, but in some cases the method was used to find what other rates would be acceptable (not necessarily all rates above the minimum) This was first applied to sampling...

Ngày tải lên: 18/06/2014, 10:05

... − ␣ ) y sinc ␣ y = sin X sin Y sin Y XY 1Y [Y = ␣ Y, Y = (1 − ␣ ) Y ] 106 Fourier Transforms in Radar and Signal Processing Figure 5.14 Filter weights with oversampling and raised cosine rounded ... braces {} has much lower side lobes, though a wider main lobe, than the basic sinc function, as should be expected from the form of the gating, or windowing, function G (Hamming weighting) With the ... Filter weights with oversampling and trapezoidal rounded gate 103 104 Fourier Transforms in Radar and Signal Processing Figure 5.13 Raised cosine rounding and ͭ ͫ g (t ) = qF sinc qFt sinc ( q − 1)...

Ngày tải lên: 18/06/2014, 10:05

... series, providing insight into the underlying principles The first main application was to find the FIR filter weights that would provide interpolation for any band-limited signal In principle, this ... only interested in its form in the frequency interval containing significant signal energy If, as we have generally assumed, the signal is limited (after downconversion to complex baseband) to the ... substituting in (6.15), we obtain (−␲ i )r (2x )r rect (x ) ⇔ sinc(r ) ( y ) If we introduce the notation (6.16) 132 Fourier Transforms in Radar and Signal Processing sncr ( y ) = dr ␲ r dy r sinc...

Ngày tải lên: 18/06/2014, 10:05

... the case of only integer delay compensation (in units of the sampling interval) [In Figure 6.8(a), this curve is identical to the dashed curve, as all the delays are within ±0.5 sampling intervals, ... be included in the integral and hence also with S.] The Fourier transform of the power spectrum in (6.42) is [as in (5.45)] ␳ a (t ) = sinc [(1 − a ) Ft /2] sinc [(1 + a ) Ft /2] (6.43) To find ... range of problems without having to perform explicitly any integration whatever After showing that the method is successful in a single channel, including compensating for both amplitude distortion...

Ngày tải lên: 18/06/2014, 10:05

... units) from the center In this case, we not 172 Fourier Transforms in Radar and Signal Processing take the amplitudes of the sinc functions at ±1 to be 0.5, as above Thus we have, again using ... repetition interval in the u domain Then it is natural to choose I to be the interval [−U /2, U /2], which is equivalent to including the factor rect (u /U ) in the integrands in (7.34) In this case, ... linear array, again of 16 elements, for both the unweighted case (rectangular aperture weighting, dotted line) and raised cosine weighting (solid line) 168 Fourier Transforms in Radar and Signal...

Ngày tải lên: 18/06/2014, 10:05

... showing the benefits of oversampling (only by a relatively small factor in some cases) in reducing the amount of computation needed in the signal processing under consideration As computing speed ... relatively challenging problems with quite modest effort It was seen in Section 7.3.3 that very good beams covering a sector at constant gain can be produced, again very easily, using the rulesand-pairs ... are potential grating lobes, which are generally undesirable, but if the repetition interval is adequate, there will be no repetitions within the basic interval in u corresponding to real space...

Ngày tải lên: 18/06/2014, 10:05

... overlapping sinc functions These are shown as dotted lines in Figure 3.9(b), with the pulse spectrum as the solid line The 48 Fourier Transforms in Radar and Signal Processing Figure 3.9 Raised cosine ... an interesting exercise in the use of the rules-and-pairs method, showing that the method gives a solution for the spectrum quite 39 40 Fourier Transforms in Radar and Signal Processing easily ... the poor spectrum, with high side lobes This discontinuity in level is removed by generating rising and falling edges of finite slope In the case of the symmetric trapezoidal pulse, this is achieved...

Ngày tải lên: 18/06/2014, 15:20

... the information in the waveform is retained by sampling it at a rate 2W (or higher) In principle, reconstituting the waveform in this case is achieved by driving a rectangular bandwidth low-pass ... (3.30) 58 Fourier Transforms in Radar and Signal Processing As a check, we note that if we used a single rounding function r , with transform R , the expression in (3.30) reduces to R( f ) ͩ sinc f ... (k integral, ≤ ␣ < 1), then all the information in the waveform is retained by sampling it at a rate 2rW, where r is given in (4.13) above 74 Fourier Transforms in Radar and Signal Processing...

Ngày tải lên: 18/06/2014, 15:20

... that it must remain finite), as there is no spectral power in these regions Thus we are free to choose G to be of any form as long as it satisfies the conditions in (5.7) In the case of sampling ... Hilbert sampling and quadrature sampling, discussed 82 Fourier Transforms in Radar and Signal Processing above, in particular from the point of view of finding the minimum sampling rate needed ... retain the signal information, but in some cases the method was used to find what other rates would be acceptable (not necessarily all rates above the minimum) This was first applied to sampling...

Ngày tải lên: 18/06/2014, 15:20

... the case of only integer delay compensation (in units of the sampling interval) [In Figure 6.8(a), this curve is identical to the dashed curve, as all the delays are within ±0.5 sampling intervals, ... be included in the integral and hence also with S.] The Fourier transform of the power spectrum in (6.42) is [as in (5.45)] ␳ a (t ) = sinc [(1 − a ) Ft /2] sinc [(1 + a ) Ft /2] (6.43) To find ... range of problems without having to perform explicitly any integration whatever After showing that the method is successful in a single channel, including compensating for both amplitude distortion...

Ngày tải lên: 18/06/2014, 15:20

... units) from the center In this case, we not 172 Fourier Transforms in Radar and Signal Processing take the amplitudes of the sinc functions at ±1 to be 0.5, as above Thus we have, again using ... repetition interval in the u domain Then it is natural to choose I to be the interval [−U /2, U /2], which is equivalent to including the factor rect (u /U ) in the integrands in (7.34) In this case, ... linear array, again of 16 elements, for both the unweighted case (rectangular aperture weighting, dotted line) and raised cosine weighting (solid line) 168 Fourier Transforms in Radar and Signal...

Ngày tải lên: 18/06/2014, 15:20

... When Pluto was first discovered, nothing was known about its orbit or its composition Pluto has an orbit that is not in the same plane as the other planets (i.e., it is tilted) and its orbit is more ... look for scatter plots with different trends—including: I I I I I I I I I I increase decrease rapid increase, followed by leveling off slow increase, followed by rapid increase rise to a maximum, ... because it has been classified as one for more than sixty years since its discovery Scientist Pluto should no longer be classified as a planet based on new evidence that has come to light in the last...

Ngày tải lên: 18/06/2014, 17:20

... techniques for carrying out Fourier transforms of suitable functions without using integration In the rest of the book the definitions and results given here will be used to obtain useful results ... function, with peak value at y = (This convolution is shown in Figure 3.4, with A = and T = in this case.) 28 Fourier Transforms in Radar and Signal Processing ∞ ͐−∞ sinc (x − m ) sinc (x − n ... y ), and this is Rule 6b However, in this case, the result is easily obtained from the definitions of the Fourier transform in (2.2), as shown in Appendix 2B In Table 2.2, not only are pairs 1b,...

Ngày tải lên: 19/06/2014, 22:20

... overlapping sinc functions These are shown as dotted lines in Figure 3.9(b), with the pulse spectrum as the solid line The 48 Fourier Transforms in Radar and Signal Processing Figure 3.9 Raised cosine ... an interesting exercise in the use of the rules-and-pairs method, showing that the method gives a solution for the spectrum quite 39 40 Fourier Transforms in Radar and Signal Processing easily ... the poor spectrum, with high side lobes This discontinuity in level is removed by generating rising and falling edges of finite slope In the case of the symmetric trapezoidal pulse, this is achieved...

Ngày tải lên: 19/06/2014, 22:20

... the information in the waveform is retained by sampling it at a rate 2W (or higher) In principle, reconstituting the waveform in this case is achieved by driving a rectangular bandwidth low-pass ... (3.30) 58 Fourier Transforms in Radar and Signal Processing As a check, we note that if we used a single rounding function r , with transform R , the expression in (3.30) reduces to R( f ) ͩ sinc f ... (k integral, ≤ ␣ < 1), then all the information in the waveform is retained by sampling it at a rate 2rW, where r is given in (4.13) above 74 Fourier Transforms in Radar and Signal Processing...

Ngày tải lên: 19/06/2014, 22:20

... that it must remain finite), as there is no spectral power in these regions Thus we are free to choose G to be of any form as long as it satisfies the conditions in (5.7) In the case of sampling ... Hilbert sampling and quadrature sampling, discussed 82 Fourier Transforms in Radar and Signal Processing above, in particular from the point of view of finding the minimum sampling rate needed ... retain the signal information, but in some cases the method was used to find what other rates would be acceptable (not necessarily all rates above the minimum) This was first applied to sampling...

Ngày tải lên: 19/06/2014, 22:20