Radix-2 FFT Algorithm
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Questions and Answers

What is the relationship between V11(k) and V12(k) in the N/2 point DFT calculation?

  • V11(k) is equal to the average of V12(k) and V21(k).
  • V11(k) is the sum of V11(k) and a phase-modified V12(k). (correct)
  • V11(k) is equal to V12(k).
  • V11(k) is derived by adding a phase factor to V12(k).

Which operation is performed on the complex number 'b' in the butterfly computation?

  • It is squared.
  • It is halved.
  • It is reflected about the origin.
  • It is multiplied by the phase factor WNk. (correct)

How many stages are required for the computation of an 8-point DFT?

  • 4 stages
  • 1 stage
  • 3 stages (correct)
  • 2 stages

What does the term 'bit reversed order' refer to in the context of DFTs?

<p>Rearranging the indices of the sequence. (C)</p> Signup and view all the answers

What is the significance of 'N = 8' in the context of the given DFT calculations?

<p>It indicates the number of points in the input sequence. (B)</p> Signup and view all the answers

What is the next step after obtaining the N/4 point DFTs in the decimation process?

<p>Repeat the process to obtain N/2 point DFTs. (B)</p> Signup and view all the answers

Which of the following describes the butterfly computation's result?

<p>It produces two new output values A and B from inputs a and b. (C)</p> Signup and view all the answers

In the given context, what are V21(n) and V22(n) derived from?

<p>They are decimated versions of the sequence f1(n). (D)</p> Signup and view all the answers

What is the main advantage of the Radix-2 DIF FFT algorithm over the Radix-2 DIT FFT algorithm?

<p>It does not require reordering of the original sequence. (D)</p> Signup and view all the answers

How are the subsequences created in the Radix-2 DIF FFT algorithm?

<p>By dividing the original sequence into the first half and second half. (D)</p> Signup and view all the answers

What is the final sequence size reached during the decimation process in Radix-2 DIF FFT?

<p>2 (C)</p> Signup and view all the answers

What do the terms G1(k) and G2(k) represent in the Radix-2 DIF FFT process?

<p>The N/2 point sequences obtained from decimation. (C)</p> Signup and view all the answers

What is the computation method for combining two complex numbers in Radix-2 DIF FFT?

<p>Their sum and difference are computed. (C)</p> Signup and view all the answers

What forms the new complex number 'B' in the Radix-2 DIF FFT algorithm?

<p>The difference term 'a-b' multiplied by the phase factor 'W k'. (C)</p> Signup and view all the answers

In the N-point DFT, how is X(k) determined for even and odd indices?

<p>X(k) for even indices corresponds to G1(k) and odd to G2(k). (B)</p> Signup and view all the answers

What is the result of decimating the N/2 point sequence G1(k) in the Radix-2 DIF FFT?

<p>Two numbers of N/4 point sequences. (D)</p> Signup and view all the answers

What is the primary function of the Decimation in Time (DIT) Radix 2 FFT algorithm?

<p>To convert the time domain N-point sequence to a frequency domain N-point sequence. (B)</p> Signup and view all the answers

In the DIT Radix 2 FFT, how are the even and odd samples of a time domain sequence designated?

<p>Even samples comprise f1(n) and odd samples comprise f2(n). (C)</p> Signup and view all the answers

What size of DFTs does the DIT Radix 2 FFT primarily work with in each step?

<p>2-point DFTs are computed at each step until reaching N-point DFT. (B)</p> Signup and view all the answers

How is the relationship between N-point DFT X(k) and N/2 point DFTs F1(k) and F2(k) expressed?

<p>X(k) = F1(k) + Wk N F2(k) (B)</p> Signup and view all the answers

What happens to the sequences f1(n) and f2(n) after the first decimation in time?

<p>They are further decimated into N/4 point sequences. (C)</p> Signup and view all the answers

Which mathematical operation is performed on the 2-point sequences during the DIT Radix 2 FFT process?

<p>Computing the 2-point DFT for each sequence. (A)</p> Signup and view all the answers

What is represented by F1(k) in the Radix 2 FFT process?

<p>The N/2 point DFT of the even-indexed samples. (C)</p> Signup and view all the answers

What mathematical summation is involved in calculating F1(k)?

<p>A summation over 2-point sequences to find the N/2 point DFT. (D)</p> Signup and view all the answers

Flashcards

Decimation in Time (DIT) Radix-2 FFT

An algorithm that converts a time-domain sequence into a frequency-domain sequence by recursively dividing the input sequence into smaller sequences and calculating their DFTs.

N-point DFT

Discrete Fourier Transform of an N-sample sequence.

Decimation in time

A process in the DIT FFT where the input sequence is broken down into smaller subsequences.

2-point DFT

Discrete Fourier Transform computed on a sequence of two samples.

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Radix-2 FFT

A fast Fourier transform algorithm based on decomposing the DFT into smaller, 2-point DFT calculations.

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x(n)

N-sample time-domain sequence.

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f1(n)

Subsequence of even-indexed samples of x(n).

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X(k)

N-point DFT of x(n).

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DIF FFT

A type of Fast Fourier Transform (FFT) algorithm where the input sequence is decomposed into smaller subsequences in the frequency domain.

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Butterfly computation

A core operation in DIF FFT that combines two complex numbers to produce a new complex number with its sum and another with its difference, multiplied by a specific phase factor.

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Phase factor

A complex number used in the butterfly computation to introduce the necessary phase shift for the frequency domain representation.

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What is the difference between DIF and DIT?

DIF FFT splits the input sequence in the frequency domain, while DIT FFT splits it in the time domain. DIT requires reordering the input sequence.

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How does DIF FFT work?

DIF FFT iteratively breaks down an N-point sequence into two N/2-point sequences, then into two N/4-point sequences, and so on, until we get 2-point sequences. This process repeats until we have N/2 numbers of 2-point sequences.

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What does G1(k) represent?

G1(k) represents the N/2 point DFT of the subsequence g1(n) obtained after the first decimation stage in DIF FFT.

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What does D11(k) represent?

D11(k) represents the N/4 point DFT of the subsequence obtained from further decimation of the N/2 point sequence G1(k).

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What is the purpose of the decimation process in DIF FFT?

Decimation in DIF FFT recursively splits the frequency domain representation into smaller subsequences, allowing for efficient computation of the DFT through smaller, more manageable DFTs.

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Radix-2 DIT FFT

A Fast Fourier Transform (FFT) algorithm that breaks down an N-point DFT into smaller DFTs of size N/2, N/4, ... until reaching 2-point DFTs. It recursively divides the input sequence based on the even and odd samples.

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Decimation in Time (DIT)

A technique in FFT algorithms where the input sequence is broken down into smaller subsequences by repeatedly dividing it in half based on even and odd indices. This process leads to smaller DFTs that can be efficiently computed.

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f1(n) and f2(n)

Subsequences derived from the original sequence 'x(n)' by taking even and odd samples, respectively. These sequences are used in DIT FFT to calculate smaller DFTs.

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V11(n), V12(n), V21(n), V22(n)

These are N/4-point sequences obtained by decimating the even and odd samples of f1(n) and f2(n) respectively. They are used in DIT FFT to perform the smaller DFT calculations.

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WkN/2

The twiddle factor used in FFT algorithms to introduce phase shifts. It's calculated as e^(-2pii*k/N) where 'k' is the frequency index and 'N' is the sequence length.

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Study Notes

Decimation in Time (DIT) Radix-2 FFT

  • Converts N-point time-domain sequence x(n) to N-point frequency-domain sequence X(k)
  • Time-domain sequence x(n) is broken down into smaller point DFTs
  • Results of smaller DFTs are combined to get the final N-point DFT
  • Radix-2 FFT: Decimation process involves breaking the sequence into smaller sub-sequences (e.g., 2-point, 4-point, 8-point DFTs) until 2-point sequences are reached. This recursive approach continues until the N-point DFT is computed.

Decimation in Time Algorithm Details

  • The N-point DFT can be computed using N/2 two-point DFTs.
  • An N/2 point DFT can be computed by two N/4 point DFTs, and so on.
  • A sequence of length N can be divided into two sequences of length N/2.
  • Even-indexed elements form one sequence, odd-indexed elements form the other sequence.
  • The FFT algorithm is recursive, continually splitting the sequence into half recursively until it reaches 2 samples.

DFT Relationships

  • Let x(n) be an N-sample sequence.
  • f1(n) = x(2n), for n=0, 1, ...,(N/2-1)
  • f2(n) = x(2n+1), for n=0, 1, ...,(N/2-1)
  • X(k) = N-point DFT of x(n)
  • F1(k) = N/2-point DFT of f1(n)
  • F2(k) = N/2-point DFT of f2(n)
  • X(k) = F1(k) + [W^k_(N/2)] F2(k), for k=0, 1,...,N-1

Radix-2 DIT FFT Butterfly Computation

  • The algorithm involves a process called "butterfly computation".
  • In each step, two complex numbers (a and b) are involved.
  • A complex number 'b' is multiplied by a phase factor (W^nk_N).
  • The result is added or subtracted from the complex number 'a'.

Bit Reversal

  • The input sequence x(n) needs to be reordered (arranged in bit-reversed order) before the decimation process. This ensures proper data arrangement for the recursive calculations.

Decimation in Frequency (DIF) Radix-2 FFT Algorithm

  • The original sequence doesn't need reordering.
  • The sequence is divided into two parts (higher and lower halves).
  • Each half is then further divided into two sub-sequences recursively until smaller 2-point DFTs are calculated.
  • The results of 2-point DFTs are combined using a formula.

Flow Graphs for FFT Algorithms

  • Visual representations outlining the recursive computations and data flow in the radix-2 FFT algorithms (DIT and DIF).

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Description

This quiz covers the Radix-2 Fast Fourier Transform (FFT) algorithm, which converts time-domain sequences into frequency-domain sequences. It explains the decimation process, the recursive breakdown of sequences, and the relationships involved in discrete Fourier transforms. Test your knowledge on this essential signal processing technique.

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