Role of Cyclic Prefix in OFDM

Role of Cyclic Prefix in OFDM

The simple frequency-domain equalizer is possible only if the channel performs circular convolution. However, in practice, all wireless channels perform linear convolution.

This linear convolution is converted into circular convolution by adding a cyclic prefix (CP) in the OFDM architecture. The cyclic prefix makes the linear convolution imparted by the channel appear as circular convolution to the DFT process at the receiver.

Demonstration Concept

To demonstrate this, consider an OFDM signal s[n] of length 8 and a channel impulse response h[n] of length 3. When s[n] is convolved with h[n]:

• Linear convolution and circular convolution give different results.
• This mismatch causes interference in OFDM systems.

Adding Cyclic Prefix

A cyclic prefix is added by copying the last NCP samples of the OFDM symbol and appending them to the beginning. If the cyclic prefix length is at least equal to the channel delay spread, the following occurs:

• Inter-symbol interference is confined to the cyclic prefix.
• The useful received signal behaves as circular convolution.

Receiver Processing

At the receiver, the cyclic prefix samples are removed. The remaining samples correspond exactly to the circular convolution of the transmitted signal and the channel impulse response.

DFT Property

After removing the cyclic prefix, the received signal satisfies:

r[n] = IDFT { H[k] · S[k] }

This confirms that circular convolution in time domain corresponds to simple multiplication in the frequency domain.

Why OFDM needs circular convolution

In OFDM:

• Each OFDM symbol is processed using an N-point DFT
• DFT diagonalizes circular convolution, not linear convolution

So without CP:

DFT{x[n] * h[n]} ≠ X[k] H[k]

Subcarriers interfere → ICI + ISI

What the cyclic prefix actually does

The cyclic prefix:

• Copies the last L samples of the OFDM symbol
• Appends them to the front of the symbol

Original OFDM symbol:
[x0 x1 x2 ... x(N−1)]

After CP:
[x(N−L) ... x(N−1) | x0 x1 ... x(N−1)]

Where L ≥ channel delay spread

Mathematical proof

Let:

• x[n] be an N-sample OFDM symbol
• CP length L ≥ Lh - 1

Then after CP removal:

y[n] = x[n] ⊛ h[n]

Taking DFT:

Y[k] = X[k] H[k]

Equalization becomes:

ĤX[k] = Y[k] / H[k]

One complex multiplication per subcarrier

Practical example (LTE / Wi-Fi numbers)

• FFT size = 2048
• CP ≈ 4.7 µs
• Channel delay spread ≈ 1–3 µs

Because CP length > delay spread:

• No ISI
• No ICI
• Simple frequency-domain equalizer works

Without CP:

• Symbols overlap
• Subcarriers mix
• Equalizer becomes very complex

Physical interpretation

Think of CP as a guard interval in time, not frequency:

• Absorbs multipath echoes
• Makes each OFDM symbol appear periodic
• DFT “sees” a periodic extension → circular convolution

What happens if CP is too short?

If CP length < channel delay spread:

• Linear convolution leaks into FFT window
• Circularity breaks
• ISI + ICI appear
• Orthogonality is lost

Summary

By adding a cyclic prefix, OFDM systems convert linear channel convolution into circular convolution, enabling low-complexity frequency-domain equalization and eliminating inter-symbol interference.

Cyclic prefix converts linear convolution into circular convolution, so that the FFT sees a diagonal frequency response, allowing simple frequency-domain equalization.

Further Reading

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