Inverse DFT (IDFT) Using Radix-2 FFT 

8 Short Question Answers & 5 Worked Out Examples on 4-Point & 8-Point Inverse DFT

This page explains the Inverse Discrete Fourier Transform (IDFT) using the Fast Fourier Transform (FFT) algorithm. It begins by comparing the computational efficiency of DFT and FFT, showing how FFT significantly reduces calculations. The core of the explanation focuses on the IDFT process using the radix-2 FFT method, highlighting the similarities and key differences compared to the forward FFT. A practical example of calculating the response of a Linear Time-Invariant (LTI) system using the IDFT via FFT is provided, demonstrating the application of the discussed concepts. Finally, it emphasizes the importance of converting linear convolution to circular convolution for efficient computation using FFT.

In this video, we explain how to compute the Inverse Discrete Fourier Transform (IDFT) using the Radix-2 FFT algorithm. We demonstrate the process of converting frequency-domain data back into the time domain, which is essential in digital signal processing (DSP) and signal analysis. This tutorial is designed for students in electrical, electronics, communications, and computer science engineering, especially those studying signals and systems. Learn how N-point DFT and IDFT are applied in real-world signal processing tasks. 

Comparison of DFT and FFT Calculations

• DFT: The number of complex additions is N(N-1), and the number of complex multiplications is N^2.

• FFT: The number of complex additions is N * M, where M is the number of stages and M = log₂(N). Thus, the number of complex additions is N * log₂(N). The number of complex multiplications is (N/2) * log₂(N).

• The source provides examples for N = 4, 8, and 16 to demonstrate the computational savings of FFT over DFT. For example, when N=16, DFT requires 240 complex additions and 256 complex multiplications, while FFT requires only 64 complex additions and 32 complex multiplications.

Inverse DFT using Radix-2 FFT

• The process for inverse DFT using FFT is very similar to the forward DFT using FFT. The key is to use the conjugate of the twiddle factor in the process and to divide the final output by N.

• Steps for IDFT:

  1. Bit Reversal: If using a Decimation-in-Time (DIT) algorithm, the input sequence (XK) must be in bit-reversed order. If using a Decimation-in-Frequency (DIF) algorithm, the output sequence will be in bit-reversed order.

  2. Stage-wise Computation: Perform stage-wise computations as done in FFT. The only difference is that the conjugate of the twiddle factor is used. For a 4-point sequence, there are two stages; for an 8-point sequence, there are three stages; and for a 16-point sequence, there are four stages.

  3. Output Division: Divide the resulting output sequence by N, where N is the number of samples. This division by N is part of the IDFT formula, which is 1/N * Σ[Xk * Wnk], where Wnk is the complex exponential.

Example: LTI System Response using Radix-2 FFT and IDFT

• The video uses an example to demonstrate how to find the output of a Linear Time-Invariant (LTI) system using Radix-2 FFT and IDFT.

• Linear vs. Circular Convolution: FFT performs circular convolution, not linear convolution. To obtain linear convolution using FFT, the input sequences x[n] and h[n] need to be padded with zeros to a length of N1 + N2 - 1, where N1 and N2 are the lengths of x[n] and h[n], respectively.

• Steps to Find LTI Response:

  • Convert to N-Point Sequences: Convert both the input sequence, x[n], and the impulse response, h[n], into N-point sequences by appending zeros. N is calculated as N1 + N2 - 1.

  • Compute XK and HK: Compute the FFT of x[n] to obtain XK, and compute the FFT of h[n] to obtain HK.

  • Compute YK: Multiply XK and HK to obtain YK, which is the frequency response of the output.

  • Compute YN: Compute the inverse FFT (IDFT) of YK to obtain y[n], which is the time-domain output.

Key Takeaways

• The core process of the FFT algorithm is used for both DFT and inverse DFT calculations.

• The key difference between the FFT and the inverse FFT is using the conjugate of the twiddle factor and dividing the result by N.

• To perform linear convolution using FFT, you need to pad the input sequences with zeros to the correct length.

• The example in the source shows a practical use case, calculating the output of an LTI system using FFT and inverse FFT.

Okay, here are 10 short question-and-answer pairs based on the provided sources and our conversation history:

1. Question: What is the main difference in the number of calculations between DFT and FFT? 

Answer: FFT significantly reduces the number of complex additions and multiplications compared to DFT, especially for larger N. For example, for N=16, FFT requires 64 additions and 32 multiplications while DFT requires 240 and 256 respectively.

2. Question: What is the twiddle factor and how is it used in IDFT? 

Answer: The twiddle factor (Wnk) is a complex exponential term used in both DFT and IDFT calculations. In IDFT, the conjugate of the twiddle factor is used.

3. Question: What is the first step when computing the inverse DFT using a DIT radix-2 FFT algorithm? 

Answer: The first step is to arrange the input sequence (XK) in bit-reversed order.

4. Question: What is done to the output sequence after completing the stage-wise computations in IDFT? 

Answer: The output sequence is divided by N, where N is the number of samples.

5. Question: What type of convolution does FFT perform directly? 

Answer: FFT performs circular convolution.

6. Question: How is linear convolution achieved using FFT? 

Answer: Linear convolution is achieved by padding the input sequences with zeros to a length of N1 + N2 - 1, where N1 and N2 are the lengths of the input sequences.

7. Question: What are the five steps to find the response of an LTI system using radix-2 DFT and IDFT, according to the source? 

Answer: The five steps are: 1) Convert x[n] and h[n] to N-point sequences, 2) compute XK (FFT of x[n]), 3) compute HK (FFT of h[n]), 4) compute YK = XK * HK, and 5) compute the inverse DFT of YK to get YN.

8. Question: What is the core concept that is used for both DFT and inverse DFT in the FFT? 

Answer: The core process of the FFT algorithm is used for both DFT and inverse DFT, but the key differences are using the conjugate of the twiddle factor and dividing the result by N.

5 Worked Out Examples on 4-Point & 8-Point Inverse DFT:

The Fast Fourier Transform (FFT) is an efficient algorithm used to compute the Discrete Fourier Transform (DFT) and its inverse. The DFT is a mathematical technique that transforms a sequence of values (often representing time-domain signals) into components of different frequencies, enabling analysis and manipulation in the frequency domain. The FFT significantly reduces the computational complexity of this transformation.

Key Concepts:

1. Fourier Transform:
   
   

   • Converts a signal from the time domain to the frequency domain.

   • Useful for analyzing the frequency content of a signal.

2. Discrete Fourier Transform (DFT):
   
   

   • Applies Fourier Transform to discrete data, typically sampled points of a signal.

   • Requires O(N2)O(N^2) operations for NN data points.

3. Fast Fourier Transform (FFT):
   
   

   • An optimized algorithm to compute the DFT.

   • Reduces the computational complexity from O(N2)O(N^2) to O(Nlog⁡N)O(N \log N), making it much faster, especially for large datasets.

Applications of FFT:

• Signal Processing: Filtering, compression, and analysis of audio, image, and video signals.

• Communication Systems: Analyzing and designing modulations and demodulations.

• Data Compression: Algorithms like JPEG and MP3 use FFT.

• Medical Imaging: Techniques like MRI and CT scans.

• Scientific Computing: Solving partial differential equations, analyzing vibrations, etc.

Common FFT Algorithms:

1. Cooley-Tukey Algorithm: The most common FFT implementation, which recursively splits the DFT into smaller DFTs.

2. Radix-2 FFT: A special case of the Cooley-Tukey algorithm optimized for data sizes that are powers of 2.

3. Mixed-Radix FFT: Handles data sizes that are not powers of 2.

Example Use Case:

Given a sampled signal like audio data, the FFT can decompose it into its frequency components. This allows identification of dominant frequencies (e.g., musical notes or noise components) and aids in filtering or equalizing the sound.

FFT| Flow Graph / Butterfly Diagram | Decimation in Frequency (DIF) FFT | Fast Fourier Transform | N-Point FFT/ Radix-r FFT/ Radix- 2 FFT | Flow Graph/ Butterfly Diagram 

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