OFDM Baseband to Passband Conversion: A Complete Guide

 

The main idea is that an OFDM transmitter first creates many modulated subcarriers digitally at baseband, combines them into one complex signal using an IFFT, and then converts that complex baseband signal into a real RF waveform using I/Q upconversion.

1. The overall OFDM transmitter

Input bits

0s and 1s

QAM / QPSK mapping

Bits → complex symbols $X[k]$

IFFT

Frequency-domain symbols → time-domain samples $x[n]$

Cyclic prefix + DAC

Prepare the discrete signal for transmission

I branch

$I(t)\cos(2\pi f_ct)$

Q branch

$-Q(t)\sin(2\pi f_ct)$

RF output

$s(t)=I(t)\cos(2\pi f_ct)-Q(t)\sin(2\pi f_ct)$

The transmitter has two conceptually different stages:

• OFDM generation: construct a complex baseband signal containing many orthogonal subcarriers.

• I/Q upconversion: shift that complex baseband signal to the radio carrier frequency $f_c$.

Let's examine the equations behind each stage.

2. Step 1: Modulate the data onto subcarriers

Suppose an OFDM symbol uses $N$ subcarriers. Each subcarrier carries a complex modulation symbol:

$$X[0],X[1],\ldots,X[N-1]$$

For example, with QPSK, each $X[k]$ might take one of the values

$$X[k]\in \left\{ \frac{1+j}{\sqrt2}, \frac{1-j}{\sqrt2}, \frac{-1+j}{\sqrt2}, \frac{-1-j}{\sqrt2} \right\}.$$

Each symbol has two components:

$$X[k]=A_k+jB_k$$

where $A_k$ is the in-phase component and $B_k$ is the quadrature component.

Importantly, these complex symbols are not yet the RF waveform. They represent the data assigned to the OFDM subcarriers.

For example, if $N=4$, you might have four symbols $X[0]$, $X[1]$, $X[2]$, and $X[3]$, each carrying its own data.

3. Step 2: The IFFT combines all subcarriers

This is the heart of OFDM.

The $N$ frequency-domain symbols are converted into $N$ time-domain samples using an inverse discrete Fourier transform:

$$\boxed{ x[n]=\frac{1}{\sqrt N} \sum_{k=0}^{N-1}X[k]e^{j2\pi kn/N} }$$

for $n=0,1,\ldots,N-1$.

Here:

• $X[k]$: data symbol assigned to subcarrier $k$.

• $x[n]$: complex baseband sample at time index $n$.

• $N$: number of IFFT points.

• $e^{j2\pi kn/N}$: the complex sinusoid corresponding to subcarrier $k$.

Using Euler's identity,

$$e^{j\theta}=\cos\theta+j\sin\theta,$$

we can expand the IFFT:

$$x[n]=\frac{1}{\sqrt N}\sum_{k=0}^{N-1} X[k]\left[ \cos\left(\frac{2\pi kn}{N}\right) +j\sin\left(\frac{2\pi kn}{N}\right) \right].$$

Since $X[k]=A_k+jB_k$, each subcarrier contributes to both the real and imaginary components of $x[n]$.

We can write the final result as

$$\boxed{x[n]=I[n]+jQ[n]}$$

where $I[n]$ and $Q[n]$ are the real and imaginary parts of the IFFT output.

Important: The IFFT has already combined all the subcarriers into a single complex time-domain signal. We do not need a separate RF oscillator for every subcarrier.

The continuous-time interpretation

Let the useful OFDM symbol duration be $T_u$. The subcarrier spacing is

$$\boxed{\Delta f=\frac{1}{T_u}}$$

Ignoring the cyclic prefix, the continuous-time complex baseband waveform can be written as

$$\boxed{ x(t)=\frac{1}{\sqrt N} \sum_{k=0}^{N-1}X[k]e^{j2\pi k\Delta f t} }$$

for $0\le t<T_u$.

Each term represents a different subcarrier. Because their frequencies are integer multiples of $\Delta f$, the subcarriers are orthogonal over the useful symbol interval:

$$\int_0^{T_u} e^{j2\pi(k-m)\Delta f t}\,dt = \begin{cases} T_u,&k=m\\ 0,&k\ne m. \end{cases}$$

This orthogonality allows the receiver to separate the subcarriers using an FFT, even though their spectra overlap.

4. Step 3: Separate the complex baseband into I and Q

Suppose the IFFT produces

$$x(t)=I(t)+jQ(t).$$

For example, at a particular instant, suppose

$$x(t)=0.6+j0.8.$$

Then

$$I(t)=0.6,\qquad Q(t)=0.8.$$

These are not two separate OFDM transmissions. They are the two real-valued components of the same complex waveform.

The transmitter sends both components through separate signal paths:

• The I path carries $I(t)$.

• The Q path carries $Q(t)$.

The key point is that $I(t)$ and $Q(t)$ are generated by the same IFFT. They are generally independent components of the complex data waveform, not a signal and its Hilbert transform.

In a practical transmitter, the digital samples are converted to analog waveforms using two DAC paths, followed by filtering and RF upconversion.

5. Step 4: Upconvert the complex baseband to the RF carrier

Now we reach the equation you originally asked about.

Let the RF carrier frequency be $f_c$. The transmitter uses two quadrature carriers:

$$c_I(t)=\cos(2\pi f_ct)$$

$$c_Q(t)=-\sin(2\pi f_ct).$$

The I branch is multiplied by the cosine:

$$s_I(t)=I(t)\cos(2\pi f_ct).$$

The Q branch is multiplied by the negative sine:

$$s_Q(t)=-Q(t)\sin(2\pi f_ct).$$

Adding both branches gives the transmitted real signal:

$$\boxed{ s(t)=I(t)\cos(2\pi f_ct) -Q(t)\sin(2\pi f_ct) }$$

Now substitute $I(t)+jQ(t)=x(t)$:

$$\begin{aligned} s(t) &=\Re\left\{ [I(t)+jQ(t)]e^{j2\pi f_ct} \right\}\\ &=\Re\left\{ x(t)e^{j2\pi f_ct} \right\}. \end{aligned}$$

Therefore,

$$\boxed{s(t)=\Re\{x(t)e^{j2\pi f_ct}\}}$$

This is the standard complex-baseband-to-real-passband conversion equation.

Notice that the transmitter does not need to calculate the Hilbert transform of $I(t)$ to produce $Q(t)$. Both components already exist from the complex IFFT output.

6. What does the actual transmitted OFDM waveform look like?

Let's substitute the IFFT expression directly into the upconversion equation.

We have

$$x(t)=\frac{1}{\sqrt N} \sum_{k=0}^{N-1}X[k]e^{j2\pi k\Delta f t}.$$

The transmitted waveform is

$$s(t)=\Re\{x(t)e^{j2\pi f_ct}\}.$$

Substituting,

$$s(t)=\Re\left\{ \frac{1}{\sqrt N} \sum_{k=0}^{N-1}X[k] e^{j2\pi(f_c+k\Delta f)t} \right\}.$$

Hence,

$$\boxed{ s(t)=\frac{1}{\sqrt N} \sum_{k=0}^{N-1} \Re\left\{ X[k]e^{j2\pi(f_c+k\Delta f)t} \right\} }$$

This equation gives us a very useful interpretation:

Each complex OFDM symbol is translated to its corresponding RF subcarrier frequency, and all the resulting real waveforms are added together.

For example, suppose the carrier is $f_c=2.4\text{ GHz}$, and the subcarrier spacing is $\Delta f=15\text{ kHz}$. The subcarrier frequencies are

$$f_k=f_c+k(15\text{ kHz}).$$

For subcarrier indices $k=0,1,2,3$, the frequencies are 2.400000 GHz, 2.400015 GHz, 2.400030 GHz, and 2.400045 GHz.

In a practical OFDM system, subcarrier indices may include negative frequencies relative to the baseband center, and some bins may be reserved for guard bands or a DC null. The same principle still applies.

7. Where does the cyclic prefix fit in?

Before upconversion, OFDM transmitters usually add a cyclic prefix (CP) to each time-domain OFDM symbol.

If the useful IFFT output is

$$x[0],x[1],\ldots,x[N-1],$$

a cyclic prefix of length $N_{\rm CP}$ copies the last $N_{\rm CP}$ samples to the beginning:

$$x[N-N_{\rm CP}],\ldots,x[N-1], x[0],\ldots,x[N-1].$$

The cyclic prefix helps handle multipath propagation and enables simple frequency-domain equalization when the channel delay spread is within the CP duration.

After adding the CP, the transmitter performs the digital-to-analog conversion and filtering, then applies I/Q upconversion to generate the real RF waveform. Some implementations perform equivalent operations in a different order, but the underlying signal model is the same.

Summary

The OFDM transmitter performs two conceptually separate operations:

1. IFFT: combines independently modulated subcarriers into one complex baseband signal, $x(t)=I(t)+jQ(t)$.

2. I/Q upconversion: translates that signal to RF using \[ s(t)=I(t)\cos(2\pi f_ct)-Q(t)\sin(2\pi f_ct). \]