how to generate barcode in ssrs report (a) Find the number of (complex) multiplications required to perform this convolution directly. in Software

Generate USS Code 128 in Software (a) Find the number of (complex) multiplications required to perform this convolution directly.

(a) Find the number of (complex) multiplications required to perform this convolution directly.
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( 6 ) Repeat part (a) using the overlap-add method with 1024-point radix-2 decimation-in-time FFTs to
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evaluate the convolutions.
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(a) If .r(n) is of length N = 8192, and h(n) of length L = 512, performing the convolution directly requires
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complex multiplications. (b) Using the method of overlap-add with 1024-point FFTs. the number of multiplications is as follows. Because h(n) is of length 512, we may segment x(n) into sequences .r,(n) of length N = 512 so that the 1024-point circular convolutions of h(n) with x,(n) will be the same as linear convolutions (although we could use sections of length 5 13, this does not result in any computational savings). With the length of x(n) being equal to 8 192, this means that we will have 16 sequences of length 512. Therefore, to perform the convolution, we must compute 17 DFTs and 16 inverse DFTs. In addition, we must form the products Y , ( k ) = H (k)X,(k)for i = 1, 2, . . . 16. Thus, the total number of complex multiplicalions is approximately
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which is about 4.5 percent of the number of complex multiplies necessary to perform the convolution directly.
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Speech that is sampled at a rate of I0 kHz is to be processed in real time. Part of the computations required involve collecting blocks of 1024 speech values m d computing a 1024-point DFT and a 1024point inverse DFT. If it takes I p s for each real multiply. how much time remains for processing the data after the DFT and the inverse DFT are computed With a 10-kHz sampling rate, a block of 1024 samples is collected every 102.4 ms. With a radix-2 FFT, the number of complex multiplications for a 1024-point DFT is approximately 5 12 log, 1024 = 5120. With a complex multiply consisting of four real multiphes. this means that we have to perform 5.120. 4 = 20,480 real multiplies for the DFT and the same number for the inverse DFT. With 1 ps per multiply, this will take
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which leaves 61.44 ms for any additional processing.
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Sampling a continuous-time signal x l , ( t )for I s generates a sequence of 4096 samples.
( a ) What is the highest frequency in .rl,(t)if it was sampled without aliasing
(b) If a 4096-point DFT of the sampled signal is computed, what is the frequency spacing in hertz between the DFT coefficients'
(c) Suppose that we are only interested in the DFT samples that correspond to frequencies in the range 200 5 f 5 300 Hz. How many complex multiplies are required to evaluate these values computing the DFT directly, and how many are required if a decimation-in-time FFT is used
( d ) How many frequency samples would be needed in order for the FFT algorithm to be more efficient than evaluating the DFT directly
(a) Collecting 4096 samples in I s means that Ihe sampling frequency is ,ti = 4096 Hz. If . r , ( ~ ) is to be sampled without aliasing, the sampling frequency must be a1 least twice the highest frequency in .r,(1). Therefore, l a ( / ) should have no frequencies above fi, = 2048 Hz.
( h ) With a 4096-point DFT. we are sampling X ( e l " ) at 4096 equally spaced frequencies between 0 and 2 ~ rwhich , corresponds to 4096 frequency samples over the range 0 5 ,f 5 4096 Hz. Therefore, the frequency spacing is Af = I H z .
(c) Over the frequency range from 200 to 300 Hz we have 101 DFT samples. Because it takes 4096 complex
multiplies to evaluate each DFT coefficient, the number of multiplies necessary toevaluate only these frequency samples is On the other hand, the number of multiplications required if an FFT is used is
Therefore, even though the FFT generates all of the frequency samples in the range 0 5 ,f 5 4096 Hz, it is more efficient than evaluating these 101 samples directly.
(d) An N-point FFT requires N log, N complex multiplies. and to evaluate M DFT coefficients directly requires
M . N complex multiplica;ions. Therefore, the FFT will be more efticient in finding these M samples if
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