Orthogonality in OFDM: A Deep Dive
Great question. This goes to the heart of why OFDM works at all. We proceed step by step, from intuition to rigorous mathematics, and then connect it to spectral efficiency.
1. Why overlapping subchannels don’t cause interference in OFDM
In conventional FDM, channels are separated by guard bands so their spectra do not overlap. This wastes bandwidth.
OFDM removes guard bands by allowing overlapping spectra, but designs the subcarriers so that they do not interfere at the receiver. This is achieved through orthogonality.
2. Meaning of orthogonality (signal theory)
Two signals \( s_1(t) \) and \( s_2(t) \) are orthogonal over an interval \( T \) if their inner product is zero:
\[\langle s_1, s_2 \rangle = \int_0^T s_1(t)\, s_2^*(t)\, dt = 0\]
This is analogous to orthogonal vectors in linear algebra:
\[\mathbf{v}_1 \cdot \mathbf{v}_2 = 0\]
Key idea: Signals may overlap in frequency but still be orthogonal over time.
3. OFDM subcarrier model
Each OFDM subcarrier is a complex exponential:
\[s_k(t) = e^{j2\pi f_k t}, \quad 0 \le t \le T\]
- \( f_k \): frequency of the \(k\)-th subcarrier
- \( T \): OFDM symbol duration
The transmitted OFDM signal:
\[x(t) = \sum_{k=0}^{N-1} X_k e^{j2\pi f_k t}\]
- \( X_k \): complex QAM/PSK symbol
- \( N \): number of subcarriers
4. Orthogonality condition in OFDM
Two subcarriers \(k\) and \(m\) are orthogonal if:
\[\int_0^T e^{j2\pi f_k t} e^{-j2\pi f_m t} dt = 0, \quad k \neq m\]
Simplifying:
\[\int_0^T e^{j2\pi (f_k - f_m)t} dt\]
This equals zero if and only if:
\[f_k - f_m = \frac{n}{T}, \quad n \in \mathbb{Z}, n \neq 0\]
OFDM design rule
\[\boxed{\Delta f = \frac{1}{T}}\]
Subcarrier spacing equals the reciprocal of the symbol duration.
5. Why overlapping spectra remain orthogonal
Each subcarrier has a sinc-shaped spectrum:
\[S_k(f) = T \cdot \text{sinc}\big(T(f - f_k)\big)\]
Property:
\[\text{sinc}(n) = 0 \quad \text{for all integers } n \neq 0\]
- Each subcarrier peak lies at the zero crossings of all others
- Spectra overlap, but sampled interference is zero
This is spectral orthogonality, not spectral separation.
6. Receiver perspective (FFT interpretation)
Demodulation of subcarrier \(m\):
\[\hat{X}_m = \int_0^T x(t) e^{-j2\pi f_m t} dt\]
Substituting \(x(t)\):
\[\hat{X}_m = \sum_{k=0}^{N-1} X_k \int_0^T e^{j2\pi (f_k - f_m)t} dt\]
Using orthogonality:
\[\hat{X}_m = \begin{cases} X_m T, & k = m \\ 0, & k \neq m \end{cases}\]
\[\boxed{\text{No Inter-Carrier Interference (ICI)}}\]
7. Discrete-time (DFT/FFT) view
Transmitter
\[x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{j2\pi kn/N}\]
Receiver
\[X_k = \sum_{n=0}^{N-1} x[n] e^{-j2\pi kn/N}\]
Orthogonality:
\[\sum_{n=0}^{N-1} e^{j2\pi (k-m)n/N} = \begin{cases} N, & k = m \\ 0, & k \neq m \end{cases}\]
8. Spectral efficiency
\[\eta = \frac{\text{bits/s}}{\text{Hz}}\]
- Overlapping subcarriers
- No guard bands
- Symbol rate per subcarrier = \(1/T\)
- Total bandwidth ≈ \(N/T\)
\[\eta \approx \log_2(M) \quad \text{bits/s/Hz (M-QAM)}\]
9. What breaks orthogonality
- Time offset
- Frequency offset / Doppler
- Multipath delay spread
Solution: Cyclic Prefix (CP) preserves orthogonality by enforcing circular convolution.
10. Summary
Orthogonality in OFDM means that each subcarrier is mathematically invisible to all others over one symbol duration, despite occupying the same spectrum.
- Not non-overlapping
- Not independent
- Linearly orthogonal basis functions
Orthogonality of Sine and Cosine: A Practical Example
Let’s explain this with a concrete, practical example, no abstraction, and with numbers you can visualize.
We will show exactly how sine and cosine cancel in practice and why the angular-frequency spacing
\[\Delta \omega = \frac{2\pi}{T}\]
makes them orthogonal.
1. What “orthogonal” means in practice
Two signals are orthogonal over a time interval \(T\) if:
\[\int_0^T x(t)\,y(t)\,dt = 0\]
Physically, this means:
- Positive and negative areas cancel
- The receiver sees zero correlation
- One signal contributes no energy to the other
2. Choose a real, practical time interval
Let’s choose:
\[T = 1\ \text{ms}\]
This could represent:
- An FFT window
- An OFDM symbol
- A DSP measurement interval
3. Compute the angular-frequency spacing
Using the rule:
\[\Delta \omega = \frac{2\pi}{T}\]
\[\Delta \omega = \frac{2\pi}{0.001} = 2000\pi\ \text{rad/s}\]
4. Define practical sine and cosine signals
Choose:
\[x(t) = \sin(\omega_1 t)\]
\[y(t) = \cos(\omega_2 t)\]
with angular frequencies spaced by \(\Delta\omega\):
\[\omega_1 = 2000\pi\ \text{rad/s}\]
\[\omega_2 = 4000\pi\ \text{rad/s}\]
These correspond to:
- 1 kHz sine wave
- 2 kHz cosine wave
5. Compute their inner product
Evaluate:
\[\int_0^{1\text{ ms}} \sin(2000\pi t)\cos(4000\pi t)\,dt\]
Using the identity:
\[\sin A \cos B = \frac{1}{2}\big[\sin(A+B) + \sin(A-B)\big]\]
We get:
\[\frac{1}{2} \int_0^{1\text{ ms}} \big[ \sin(6000\pi t) + \sin(-2000\pi t) \big] dt\]
Each sine term completes an integer number of half-periods over \(T\):
\[\int_0^T \sin\left(n\frac{2\pi}{T}t\right) dt = 0\]
Hence:
\[\boxed{ \int_0^{1\text{ ms}} \sin(2000\pi t)\cos(4000\pi t)\,dt = 0 }\]
6. What is happening physically?
- The 1 kHz sine completes 1 full cycle
- The 2 kHz cosine completes 2 full cycles
- Positive and negative lobes perfectly balance
When you “multiply and average” (as correlators and FFTs do), the result is zero. This is orthogonality in action.
7. What if spacing is NOT \(2\pi/T\)?
- 1 kHz sine
- 1.5 kHz cosine
Angular spacing ≠ 2Ï€/T. Result:
- Partial cycles
- Incomplete cancellation
- Integral ≠ 0 → interference
This is exactly what causes spectral leakage and ICI in FFT/OFDM systems.
8. Connection to FFT and OFDM
- FFT window length = \(T\)
- FFT bin spacing = 1/T (Hz)
- Angular spacing = 2Ï€/T
Each bin corresponds to:
\[\omega_k = k\frac{2\pi}{T}\]
Therefore:
- Sine and cosine basis functions are orthogonal
- FFT perfectly separates frequencies
- OFDM subcarriers do not interfere
9. Intuitive picture
Think in terms of phase rotation. Over time \(T\), phase advances by:
\[\Delta \phi = \omega T\]
When:
\[\omega = k\frac{2\pi}{T}\]
The phase completes an integer number of full rotations, forming a closed loop, and the average becomes zero.