acompute the rectangular form of the 3point DFT of the follo

(a)compute the rectangular form of the 3-point DFT of the following signals. x [0]=1,x [1]=1,x [2]=2, all others are zero for n <0 and n >3.

(b)
calculate IDFT of (a) (show work)

Solution

Ans-

One of the most important properties of the DTFT is the convolution property: y[n] = h[n] x[n] DTFT Y() = H() X (). This property is useful for analyzing linear systems (and for filter design), and also useful for “on paper” convolutions of two sequences h[n] and x[n], since if the sequences are simple ones whose DTFTs are known or are easily determined, we can simply multiply the two transforms and then “look up” the inverse transform to get the convolution. What if we want to automate this procedure using a computer? Right away there is a problem since is a continuous variable that runs from to , so it looks like we need an (uncountably) infinite number of ’s which cannot be done on a computer. For example, we cannot implement the ideal lowpass filter digitally. This chapter exploit what happens if we do not use all the ’s, but rather just a finite set (which can be stored digitally). In general this will entail irrecoverable information loss. Fortunately, not always though! (Otherwise DSP would be a more academic subject.) Any signal that is stored in a computer must be a finite length sequence, say x[0], x[1], . . . , x[L 1] . Since there are only L signal time samples, it stands to reason that we should not need an infinite number of frequencies to adequately represent the signal. In fact, exactly N L frequencies should be enough information. (We will see when we discuss zero-padding that for some purposes N 2L is an appropriate number of frequencies.) Main points • By the end of Chapter 5, we will know (among other things) how to use the DFT to convolve two generic sampled signals stored in a computer. By the end of Ch. 6, we will know that by using the FFT, this approach to convolution is generally much faster than using direct convolution, such as MATLAB’s conv command. • Using the DFT via the FFT lets us do a FT (of a finite length signal) to examine signal frequency content. (This is how digital spectrum analyzers work.) Chapter 3 and 4 especially focussed on DT systems. Now we focus on DT signals for a while. The discrete Fourier transform or DFT is the transform that deals with a finite discrete-time signal and a finite or discrete number of frequencies. Which frequencies? k = 2 N k, k = 0, 1, . . . , N 1. For a signal that is time-limited to 0, 1, . . . ,L 1, the above N L frequencies contain all the information in the signal, i.e., we can recover x[n] from X 2 N k N1 k=0 . However, it is also useful to see what happens if we throw away all but those N frequencies even for general aperiodic signals. Discrete-time Fourier transform (DTFT) review Recall that for a general aperiodic signal x[n], the DTFT and its inverse is X () = X n= x[n] e n , x[n] = 1 2 Z X () e n d . Discrete-time Fourier series (DTFS) review Recall that for a N-periodic signal x[n], x[n] = N X1 k=0 ck e 2 N kn where ck = 1 N N X1 n=0 x[n] e 2 N kn . 5.4 c J. Fessler, May 27, 2004, 13:14 (student version) Definition(s) The N-point DFT of any signal x[n] is defined as follows: X[k] 4 = PN1 n=0 x[n] e 2 N kn , k = 0, . . . , N 1 ?? otherwise. Almost all books agree on the top part of this definition. (An exception is the 206 textbook (DSP First), which includes a 1 N out front to make the DFT match the DTFS.) But there are several possible choices for the “??” part of this definition. 1. Treat X[k] as an N-periodic function that is defined for all integer arguments k Z. This is reasonable mathematically since X[n + N] = N X1 n=0 x[n] e 2 N (k+N)n = N X1 n=0 x[n] e ( 2 N kn+2kn) = N X1 n=0 x[n] e 2 N kn = X[k] . 2. Treat X[k] as undefined for k / {0, . . . , N 1}. This is reasonable from a practical perspective since in a computer we have subroutines that take an N-point signal x[n] and return only the N values X[0], . . . , X[N 1], so trying to evaluate an expression like “X[k]” will cause an error in a computer. 3. Treat X[k] as being zero for k / {0, . . . , N 1}. This is a variation on the previous option