The chirp transform of a vector a (a 0 , a 1 , ,a n 1 ) is the vector y (y 0 , y 1 , ,y n 1 ), where y k n 1 j 0 aj z kj and z is any complex number The DFT is therefore a special case of the chirp transform, obtained by taking z n Show how to evaluate the chirp transform in time O(nlgn) for any complex number Use the equation to view the chirp transform as a convolution n 1 Yk zk2 2 (k j) 2 j 0

Question: The chirp transform of a vector a = (a 0 , a 1 , . . . ,a n - 1 ) is the vector

The chirp transform of a vector a = (a₀, a₁, . . . ,a_n_-₁) is the vector y= (y₀, y₁, . . . ,y_n_-₁), where y_k = Σ^n-1_j=0?aj z^kj and z is any complex number. The DFT is therefore a special case of the chirp transform, obtained by taking z = ω_n. Show how to evaluate the chirp transform in time?O(nlgn)?for any complex number?. Use the equation

n-1 Yk = zk2/2 -(k-j)?/2 j=0

to view the chirp transform as a convolution.

n-1 Yk = zk2/2 -(k-j)?/2 j=0

Step by Step Solution

★★★★★

3.49 Rating (166 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

The chirp transform of a vector a can be viewed as a convolution using the equation yk ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Introduction to Algorithms Questions!

A definite advantage of the FFT is that it reduces considerably the computation in the convolution sum. Thus if x[n], 0 ¤ n ¤ N 1, is the input of an FIR filter with impulse response...

We have seen how to evaluate a polynomial of degree-bound n at a single point in O(n) time using Horner's rule. We have also discovered how to evaluate such a polynomial at all n complex roots of...

The definition of the conditional p.d.f. of X given Y = y is arbitrary if f2(y) = 0. The reason that this causes no serious problem is that it is highly unlikely that we will observe Y close to a...

PYTHON PROGRAMMING . internal form: Graphics is a core foundation of computing technology. Although it is common to hear this word in gaming contexts, graphics can be seen in every aspect of using a...

Consider an economy that has three possible states in period 1: a, b and c. There are two securities in the market, security 1 and security 2, with the following payment vectors: X1 = [1; 1; 1) and...

13 12 The unit tangent vector T and the principal unit normal vector N for the parameterized curve r(t) = 3 2 (, t> 0, are shown below. Use the definitions to compute the unit binormal vector B and...

1 point Given that the acceleration vector is a t 1 cos t 1 sin t 2t the initial velocity is v 0 1 0 1 and the initial position vector is r 0 1 1 1 compute A The velocity vector v t B The position...

1 3- ) 00:15 4 M. 101 44.0 ... ... 38 . Addition, Multiplication o Time Shifting o Time inversion o Time Scaling o Amplitude Scaling Software: Matlab Lab Tasks: T- Generation and Plotting of Basic...

Write a function named shift with arguments: x: a vector of any length. right: a length one logical vector with a default value of TRUE. The function should return a vector, the same type and...

Can someone help me with my code? Ive tried multiple different solutions with no avail. manufacturing standards for sale. workspace. Make sure the barData.mat file is in the same folder as your...

What is the osmotic pressure formed by dissolving 44.2 mg of aspirin (C9H8O4) in 0.358 L of water at 25 oC?

Suppose that A is some unknown invertible matrix, but you know of a sequence of elementary row operations that produces the identity matrix when applied in succession to A. Explain how you can use...

Which of the following pairs of events are mutually exclusive? (a) A golfer scoring the lowest 18-hole round in a 72- hole tournament and losing the tournament. (b) A poker player getting a flush...

Summarize the points in both reports that you could fairly emphasize if asked to write a report for the press in Essex.

Assume Alice, the client, creates a TCP connection with Bob, the server. They exchange data and close the connection. Now Alice starts a new connection with Bob by sending a new SYN segment. The...

UDP is a message-oriented protocol. TCP is a byte-oriented protocol. If an application needs to protect the boundaries of its message, which protocol should be used, UDP or TCP?

We used the terms passive open and active open in discussing a connectionoriented communication using TCP. Assume there is a telephone conversation between Alice and Bob. Since a telephone...

(Time allowance estimation: 7 minutes) You have won a swepstake that gives you 12 annual installments of $25,000 with the first one to be received one year from today. If the annual discounting rate...

Shares A and B have the following returns: Stock A Stock B 1 0.09 0.05 2 0.07 0.03 3 0.13 0.04 4 0.02 0.01 5 0.07 0.02 a. What are the expected returns of the two shares? b. What are the standard...

o se that the citizens of Smalltown rank alternatives W,X,Y and Z as shown in the table from best a) What is Smalltown's raun. g over each pair of alternatives if we use the "method of majority...