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?

Rearranging the indices of the sequence. Signup and view all the answers

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

It indicates the number of points in the input sequence. Signup and view all the answers

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

Repeat the process to obtain N/2 point DFTs. Signup and view all the answers

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

It produces two new output values A and B from inputs a and b. Signup and view all the answers

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

They are decimated versions of the sequence f1(n). 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?

It does not require reordering of the original sequence. Signup and view all the answers

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

By dividing the original sequence into the first half and second half. Signup and view all the answers

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

2 Signup and view all the answers

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

The N/2 point sequences obtained from decimation. Signup and view all the answers

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

Their sum and difference are computed. Signup and view all the answers

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

The difference term 'a-b' multiplied by the phase factor 'W k'. Signup and view all the answers

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

X(k) for even indices corresponds to G1(k) and odd to G2(k). 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?

Two numbers of N/4 point sequences. Signup and view all the answers

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

To convert the time domain N-point sequence to a frequency domain N-point sequence. 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?

Even samples comprise f1(n) and odd samples comprise f2(n). Signup and view all the answers

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

2-point DFTs are computed at each step until reaching N-point DFT. 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?

X(k) = F1(k) + Wk N F2(k) Signup and view all the answers

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

They are further decimated into N/4 point sequences. Signup and view all the answers

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

Computing the 2-point DFT for each sequence. Signup and view all the answers

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

The N/2 point DFT of the even-indexed samples. Signup and view all the answers

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

A summation over 2-point sequences to find the N/2 point DFT. Signup and view all the answers

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.