Linear vs Circular Convolution

Linear convolution vs circular convolution

Linear convolution (what nature does)

If:

• x[n] = transmitted signal
• h[n] = channel impulse response (multipath)

Then the received signal is:

y[n] = x[n] * h[n]

This means:

• Past symbols spill into future symbols
• Causes inter-symbol interference (ISI)
• DFT of y[n] is not a simple product

Circular convolution (what DFT assumes)

The DFT assumes periodic signals:

y[n] = x[n] ⊛ h[n], where indices wrap around modulo N

Only under circular convolution does the DFT satisfy:

Y[k] = X[k] · H[k]

This is what enables one-tap frequency-domain equalization.

What happens in the real channel (linear convolution)

The channel performs linear convolution:

y[n] = Σ_l=0^Lh−1 h[l] x[n-l]

But because:

• CP absorbs the “spillover”
• Receiver discards CP before DFT

The useful part of the received symbol becomes:

y[n] = Σ_l=0^Lh−1 h[l] x[(n-l) mod N]

Modulo indexing = circular convolution

Linear vs Circular Convolution Deep Dive

Linear convolution example

Let:

x[n] = [1, 2, 3]

h[n] = [4, 5]

Compute linear convolution:

y[0] = 1*4 = 4
y[1] = 1*5 + 2*4 = 13
y[2] = 2*5 + 3*4 = 22
y[3] = 3*5 = 15

y[n] = [4, 13, 22, 15]

Length = 3 + 2 - 1 = 4 → output longer than input

Circular convolution example

x[n] = [1, 2, 3], h[n] = [4, 5, 0] (padded to length 3)

y[0] = 1*4 + 2*0 + 3*5 = 19
y[1] = 1*5 + 2*4 + 3*0 = 13
y[2] = 1*0 + 2*5 + 3*4 = 22

y[n] = [19, 13, 22]

Summary of differences

Feature	Linear Convolution	Circular Convolution
Signal length	N+M-1	N
Wrap-around	No	Yes (mod N)
DFT property	No simple product	Y[k] = X[k] · H[k]
Natural / artificial	Nature (channel)	FFT assumption
OFDM role	Causes ISI if unchecked	Enables one-tap equalization

Further Reading

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