Understanding IFFT Frequency Spacing
If you gave the IFFT a list of individual ‘notes’ (subcarrier amplitudes) at specific ‘pitches’ (frequencies: 0, f, 2f, 3f). The IFFT's job is to combine them into one single ‘melody’ (the time-domain signal).
Even though it's one melody, the IFFT guarantees that the individual notes are still distinctly present and perfectly spaced in frequency within that melody. We prove this because if you ‘un-mix’ the melody using the FFT, you get back exactly the original list of individual notes you started with. This means the IFFT effectively ‘defined’ and ‘placed’ them in their frequency slots within the combined signal.
The entire process demonstrates that the IFFT doesn't just randomly scramble data. Instead:
- It takes data defined by frequency slots (X array where each element is a subcarrier).
Visualizing the Concept
Here's a conceptual image to help illustrate the idea of individual notes combining into a single melody while retaining their distinct frequency presence.
That's a very good question for a student, as it goes to the heart of how the Discrete Fourier Transform (DFT) and its fast implementations (FFT/IFFT) interpret frequency.
When we talk about N points in a DFT/IFFT (in our MATLAB example, N=4), the transform inherently deals with N discrete frequency components. These frequencies are always relative to the sampling rate of the time-domain signal and are conventionally ordered.
>> ifft([1, -1, 1, 1])
ans =
0.5000 + 0.0000i 0.0000 - 0.5000i 0.5000 + 0.0000i 0.0000 + 0.5000i
>> fft(ans)
ans =
1 -1 1 1
Let's break down why it's 0, f, 2f, 3f (and what f represents):
1. The "Fundamental Frequency Step" (Δf)
Imagine our time-domain OFDM symbol has a duration of Tsymbol (excluding the CP for simplicity in this context).
If you have N samples in this duration, the inverse of that duration, 1/Tsymbol, represents the smallest frequency separation between our orthogonal subcarriers.
This smallest frequency separation is often called the subcarrier spacing or fundamental frequency step. Let's denote this as Δf.
- Δf = 1 / (N × Tsample), where Tsample is the duration of one time-domain sample.
- Or more simply, Δf = 1 / TIFFT, where TIFFT is the duration of the IFFT output (the useful part of the OFDM symbol).
2. The Subcarrier Indices (k) and Their Frequencies
The N input values to the IFFT (X[0], X[1], X[2], X[3]) correspond to specific harmonic frequencies of this Δf.
-
X[0] corresponds to k=0: This is the DC component (Direct Current), meaning it represents a frequency of 0 Hz. It's a constant (no oscillation).
Analogy: The lowest, continuous hum in the orchestra.
-
X[1] corresponds to k=1: This is the first positive frequency component. Its frequency is 1 × Î”f.
Analogy: The lowest specific note (e.g., C3).
-
X[2] corresponds to k=2: This is the second positive frequency component. Its frequency is 2 × Î”f.
Analogy: A note exactly one octave higher than 1 × Î”f (e.g., C4). This is also often referred to as the Nyquist frequency or folding frequency, especially when dealing with real-valued time signals.
-
X[3] corresponds to k=3: This is the third component. Its frequency is 3 × Î”f. However, in a standard N-point FFT/IFFT convention, X[k] for k > N/2 actually corresponds to negative frequencies.
- For N=4, X[3] is often considered to correspond to -1 × Î”f. This makes the frequencies symmetrical around DC: [0 × Î”f, 1 × Î”f, 2 × Î”f, -1 × Î”f].
- This "negative frequency" representation is particularly important when the time-domain signal is purely real. If x is real, then X[k] must be the complex conjugate of X[N-k]. So, X[3] would be the conjugate of X[1].
In your example X = [1, -1, 1, 1] for N=4:
- X[0] = 1: This is the DC component (0 Hz).
- X[1] = -1: This is the subcarrier at 1 × Î”f.
- X[2] = 1: This is the subcarrier at 2 × Î”f (Nyquist frequency).
- X[3] = 1: This is the subcarrier at -1 × Î”f.
Notice that for X[1] and X[3] (the positive Δf and negative Δf components), their values are -1 and 1. They are not complex conjugates of each other, which means the resulting time-domain signal from ifft([1, -1, 1, 1]) is indeed complex-valued, as you correctly observed in your MATLAB output: 0.5000 + 0.0000i 0.0000 - 0.5000i 0.5000 + 0.0000i 0.0000 + 0.5000i.
Why the "Spacing" is Important:
- Orthogonality: The key reason for using these integer multiples of Δf (0, 1Δf, 2Δf, 3Δf, etc.) is that these specific frequencies are mathematically orthogonal over the duration of the IFFT symbol. This means that when you combine them, and then later analyze the combined signal, you can perfectly separate them without them interfering with each other.
- Bandwidth Efficiency: By using these precisely spaced, orthogonal frequencies, OFDM can pack a lot of data into a given bandwidth very efficiently, as the subcarriers can overlap spectrally without causing inter-carrier interference (ICI) due to their orthogonality.
So, 0, f, 2f, 3f (where f is Δf) represents the intrinsic set of discrete, orthogonal frequencies that an N-point DFT/IFFT naturally works with, ensuring perfect separability of data carried on each.
The IFFT takes a sequence of frequency-domain samples (often called subcarriers or coefficients, typically complex numbers) and converts them into a time-domain signal. It's fundamentally an inverse operation to the FFT (Fast Fourier Transform).
- Input: A series of discrete values, each corresponding to the amplitude and phase of a specific frequency component (a "subcarrier") at a distinct frequency bin. These bins are inherently equally spaced in the frequency domain.
- Output: A series of discrete values representing the signal's amplitude over time.
The IFFT assumes your input data (e.g., your X array) already represents individual frequency components (subcarriers) that are perfectly spaced in the frequency domain. Its job is to synthesize a time-domain signal that, when analyzed back into the frequency domain using the FFT, will reveal those exact, perfectly spaced subcarriers. It doesn't create the spacing; it works with the inherent spacing of its frequency-domain input.