In this notebook, we will investigate the basic building blocks of an OFDM system at the transmitter and receiver side. OFDM (Orthogonal frequency division multiplexing) is a multicarrier system that is applied in a wide range of wireless transmission systems, such as LTE, WiMAX and DVB-T and DAB. The fundamental concept of a multicarrier system is the division of a high-rate transmitted data stream into several low-rate narrow subcarriers. This way, several advantages are obtained:

- Since the symbol duration is inverse proportional to the symbol rate, each subcarrier has relatively long symbols. Long symbols are robust against multipath fading, as it occurs in wireless systems.
- When a carrier is in a deep fade due to frequency-selectivity of the channel (i.e. the received energy on this carrier is very low), only the data on this subcarrier is lost, instead of the whole stream.
- Multicarrier systems allow easy multi-user resource sharing by allocating different subcarriers to different users.

Consider the following block diagram, which contains fundamental blocks for the OFDM system:

```
```

In the following OFDM example, we will go through each block and describe its operation. However, before let us define some parameters that are used for the OFDM system:

The number of subcarriers $K$ describes, how many subcarriers are available in the OFDM system.

```
K = 64 # number of OFDM subcarriers
```

```
CP = K//4 # length of the cyclic prefix: 25% of the block
```

```
P = 8 # number of pilot carriers per OFDM block
pilotValue = 3+3j # The known value each pilot transmits
```

```
allCarriers = np.arange(K) # indices of all subcarriers ([0, 1, ... K-1])
pilotCarriers = allCarriers[::K//P] # Pilots is every (K/P)th carrier.
# For convenience of channel estimation, let's make the last carriers also be a pilot
pilotCarriers = np.hstack([pilotCarriers, np.array([allCarriers[-1]])])
P = P+1
# data carriers are all remaining carriers
dataCarriers = np.delete(allCarriers, pilotCarriers)
print ("allCarriers: %s" % allCarriers)
print ("pilotCarriers: %s" % pilotCarriers)
print ("dataCarriers: %s" % dataCarriers)
plt.plot(pilotCarriers, np.zeros_like(pilotCarriers), 'bo', label='pilot')
plt.plot(dataCarriers, np.zeros_like(dataCarriers), 'ro', label='data')
```

`mapping_table`

.

```
mu = 4 # bits per symbol (i.e. 16QAM)
payloadBits_per_OFDM = len(dataCarriers)*mu # number of payload bits per OFDM symbol
mapping_table = {
(0,0,0,0) : -3-3j,
(0,0,0,1) : -3-1j,
(0,0,1,0) : -3+3j,
(0,0,1,1) : -3+1j,
(0,1,0,0) : -1-3j,
(0,1,0,1) : -1-1j,
(0,1,1,0) : -1+3j,
(0,1,1,1) : -1+1j,
(1,0,0,0) : 3-3j,
(1,0,0,1) : 3-1j,
(1,0,1,0) : 3+3j,
(1,0,1,1) : 3+1j,
(1,1,0,0) : 1-3j,
(1,1,0,1) : 1-1j,
(1,1,1,0) : 1+3j,
(1,1,1,1) : 1+1j
}
for b3 in [0, 1]:
for b2 in [0, 1]:
for b1 in [0, 1]:
for b0 in [0, 1]:
B = (b3, b2, b1, b0)
Q = mapping_table[B]
plt.plot(Q.real, Q.imag, 'bo')
plt.text(Q.real, Q.imag+0.2, "".join(str(x) for x in B), ha='center')
```

Above, we have plotted the 16QAM constellation, along with the bit-labels. Note the Gray-mapping, i.e. two adjacent constellation symbols differ only by one bit and the other 3 bits remain the same. This technique helps to minimize bit-errors, in case a wrong constellation symbol is detected: Most probably, symbol errors are "off-by-one" errors, i.e. a symbol next to the correct symbol is detected. Then, only a single bit-error occurs.

The demapping table is simply the inverse mapping of the mapping table:

```
demapping_table = {v : k for k, v in mapping_table.items()}
```

`channelResponse`

. Also, we plot the corresponding frequency response. As we see, the channel is frequency-selective. Further, we define the signal-to-noise ratio in dB, that should occur at the receiver.

```
channelResponse = np.array([1, 0, 0.3+0.3j]) # the impulse response of the wireless channel
H_exact = np.fft.fft(channelResponse, K)
plt.plot(allCarriers, abs(H_exact))
SNRdb = 25 # signal to noise-ratio in dB at the receiver
```

```
```

```
bits = np.random.binomial(n=1, p=0.5, size=(payloadBits_per_OFDM, ))
print ("Bits count: ", len(bits))
print ("First 20 bits: ", bits[:20])
print ("Mean of bits (should be around 0.5): ", np.mean(bits))
```

`bits`

are now sent to a serial-to-parallel converter, which groups the bits for the OFDM frame into a groups of $mu$ bits (i.e. one group for each subcarrier):

```
def SP(bits):
return bits.reshape((len(dataCarriers), mu))
bits_SP = SP(bits)
print ("First 5 bit groups")
print (bits_SP[:5,:])
```

`mapping_table`

.

```
def Mapping(bits):
return np.array([mapping_table[tuple(b)] for b in bits])
QAM = Mapping(bits_SP)
print ("First 5 QAM symbols and bits:")
print (bits_SP[:5,:])
print (QAM[:5])
```

`dataCarriers`

and `pilotCarriers`

. Now, to create the overall OFDM data, we need to put the data and pilots into the OFDM carriers:

```
def OFDM_symbol(QAM_payload):
symbol = np.zeros(K, dtype=complex) # the overall K subcarriers
symbol[pilotCarriers] = pilotValue # allocate the pilot subcarriers
symbol[dataCarriers] = QAM_payload # allocate the pilot subcarriers
return symbol
OFDM_data = OFDM_symbol(QAM)
print ("Number of OFDM carriers in frequency domain: ", len(OFDM_data))
```

`OFDM_data`

can be transformed to the time-domain by means of the IDFT operation.

```
def IDFT(OFDM_data):
return np.fft.ifft(OFDM_data)
OFDM_time = IDFT(OFDM_data)
print ("Number of OFDM samples in time-domain before CP: ", len(OFDM_time))
```

Subsequently, we add a cyclic prefix to the symbol. This operation concatenates a copy of the last `CP`

samples of the OFDM time domain signal to the beginning. This way, a cyclic extension is achieved. The CP fulfills two tasks:

- It isolates different OFDM blocks from each other when the wireless channel contains multiple paths, i.e. is frequency-selective.
- It turns the linear convolution with the channel into a circular one. Only with a circular convolution, we can use the single-tap equalization OFDM is so famous for.

For more information about the CP, you can refer to a dedicated article about the Cyclic Prefix in OFDM.

```
def addCP(OFDM_time):
cp = OFDM_time[-CP:] # take the last CP samples ...
return np.hstack([cp, OFDM_time]) # ... and add them to the beginning
OFDM_withCP = addCP(OFDM_time)
print ("Number of OFDM samples in time domain with CP: ", len(OFDM_withCP))
```

`channelResponse`

. Hence, the signal at the receive antenna is the convolution of the transmit signal with the channel response. Additionally, we add some noise to the signal according to the given SNR value:

```
def channel(signal):
convolved = np.convolve(signal, channelResponse)
signal_power = np.mean(abs(convolved**2))
sigma2 = signal_power * 10**(-SNRdb/10) # calculate noise power based on signal power and SNR
print ("RX Signal power: %.4f. Noise power: %.4f" % (signal_power, sigma2))
# Generate complex noise with given variance
noise = np.sqrt(sigma2/2) * (np.random.randn(*convolved.shape)+1j*np.random.randn(*convolved.shape))
return convolved + noise
OFDM_TX = OFDM_withCP
OFDM_RX = channel(OFDM_TX)
plt.figure(figsize=(8,2))
plt.plot(abs(OFDM_TX), label='TX signal')
plt.plot(abs(OFDM_RX), label='RX signal')
plt.legend(fontsize=10)
plt.xlabel('Time'); plt.ylabel('$|x(t)|$');
plt.grid(True);
```

```
def removeCP(signal):
return signal[CP:(CP+K)]
OFDM_RX_noCP = removeCP(OFDM_RX)
```

```
def DFT(OFDM_RX):
return np.fft.fft(OFDM_RX)
OFDM_demod = DFT(OFDM_RX_noCP)
```

As the next step, the wireless channel needs to be estimated. For illustration purposes, we resort to a simple zero-forcing channel estimation followed by a simple interpolation. The principle of channel estimation is as follows:

The transmit signal contains pilot values at certain pilot carriers. These pilot values and their position in the frequency domain (i.e. the pilot carrier index) are known to the receiver. From the received information at the pilot subcarriers, the receiver can estimate the effect of the wireless channel onto this subcarrier (because it knows what was transmitted and what was received). Hence, the receiver gains information about the wireless channel at the pilot carriers. However, it wants to know what happened at the data carriers. To achieve this, it interpolates the channel values between the pilot carriers to get an estimate of the channel in the data carriers.

```
def channelEstimate(OFDM_demod):
pilots = OFDM_demod[pilotCarriers] # extract the pilot values from the RX signal
Hest_at_pilots = pilots / pilotValue # divide by the transmitted pilot values
# Perform interpolation between the pilot carriers to get an estimate
# of the channel in the data carriers. Here, we interpolate absolute value and phase
# separately
Hest_abs = scipy.interpolate.interp1d(pilotCarriers, abs(Hest_at_pilots), kind='linear')(allCarriers)
Hest_phase = scipy.interpolate.interp1d(pilotCarriers, np.angle(Hest_at_pilots), kind='linear')(allCarriers)
Hest = Hest_abs * np.exp(1j*Hest_phase)
plt.plot(allCarriers, abs(H_exact), label='Correct Channel')
plt.stem(pilotCarriers, abs(Hest_at_pilots), label='Pilot estimates')
plt.plot(allCarriers, abs(Hest), label='Estimated channel via interpolation')
plt.grid(True); plt.xlabel('Carrier index'); plt.ylabel('$|H(f)|$'); plt.legend(fontsize=10)
plt.ylim(0,2)
return Hest
Hest = channelEstimate(OFDM_demod)
```

```
def equalize(OFDM_demod, Hest):
return OFDM_demod / Hest
equalized_Hest = equalize(OFDM_demod, Hest)
```

```
def get_payload(equalized):
return equalized[dataCarriers]
QAM_est = get_payload(equalized_Hest)
plt.plot(QAM_est.real, QAM_est.imag, 'bo');
```

```
def Demapping(QAM):
# array of possible constellation points
constellation = np.array([x for x in demapping_table.keys()])
# calculate distance of each RX point to each possible point
dists = abs(QAM.reshape((-1,1)) - constellation.reshape((1,-1)))
# for each element in QAM, choose the index in constellation
# that belongs to the nearest constellation point
const_index = dists.argmin(axis=1)
# get back the real constellation point
hardDecision = constellation[const_index]
# transform the constellation point into the bit groups
return np.vstack([demapping_table[C] for C in hardDecision]), hardDecision
PS_est, hardDecision = Demapping(QAM_est)
for qam, hard in zip(QAM_est, hardDecision):
plt.plot([qam.real, hard.real], [qam.imag, hard.imag], 'b-o');
plt.plot(hardDecision.real, hardDecision.imag, 'ro')
```

In the diagram above, the blue points are the received QAM points, where as the the red points connected to them are the closest possible constellation points, and the bit groups corresponding to these red points are returned.

Finally, the bit groups need to be converted to a serial stream of bits, by means of parallel to serial conversion.

```
def PS(bits):
return bits.reshape((-1,))
bits_est = PS(PS_est)
```

Now, that all bits are decoded, let's calculate the bit error rate:

```
print ("Obtained Bit error rate: ", np.sum(abs(bits-bits_est))/len(bits))
```

Do you have questions or comments? Let's dicuss below!

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