Exercises

ex-sp-ch22-01

Easy

Generate a single OFDM symbol with 64 subcarriers using QPSK data and plot the time-domain signal.

Solution
Implementation
N_fft = 64
data = np.exp(1j * np.pi * np.random.choice([1,3,5,7], N_fft) / 4)
x = np.fft.ifft(data) * np.sqrt(N_fft)

ex-sp-ch22-02

Easy

Add a cyclic prefix of length 16 to an OFDM symbol and verify that the last 16 samples equal the first 16 of the CP-extended signal.

Solution
Implementation
N_cp = 16
x_cp = np.concatenate([x[-N_cp:], x])
assert np.allclose(x_cp[:N_cp], x[-N_cp:])

ex-sp-ch22-03

Easy

Compute the PAPR of 1000 OFDM symbols with 256 subcarriers and QPSK modulation. What is the 99th percentile PAPR?

Solution
Implementation
paprs = []
for _ in range(1000):
    data = np.exp(1j*np.pi*np.random.choice([1,3,5,7],256)/4)
    x = np.fft.ifft(data)*np.sqrt(256)
    paprs.append(np.max(np.abs(x)**2)/np.mean(np.abs(x)**2))
print(f"99th percentile PAPR: {10*np.log10(np.percentile(paprs, 99)):.2f} dB")

ex-sp-ch22-04

Easy

Verify that the OFDM subcarriers are orthogonal by computing the inner product of any two different subcarriers.

Solution
Implementation
N = 64
n = np.arange(N)
for k1, k2 in [(0,1), (3,7), (10,20)]:
    ip = np.sum(np.exp(1j*2*np.pi*k1*n/N) * np.conj(np.exp(1j*2*np.pi*k2*n/N)))
    print(f"<e_{k1}, e_{k2}> = {ip:.6f}")  # should be ~0

ex-sp-ch22-05

Easy

Compute the spectral efficiency of an OFDM system with N=1024N = 1024, NCP=72N_{\text{CP}} = 72, 900 data subcarriers, and 64-QAM.

Solution
Calculation
N, Ncp, Ndata, M = 1024, 72, 900, 64
eta = (N/(N+Ncp)) * (Ndata/N) * np.log2(M)
print(f"Spectral efficiency: {eta:.2f} bps/Hz")

ex-sp-ch22-06

Medium

Build a complete OFDM transceiver with 256 subcarriers, 16-QAM, and a 5-tap Rayleigh channel. Implement pilot-based channel estimation with linear interpolation. Plot BER vs SNR.

Solution
Implementation
# See code supplement ch22/python/ofdm_system.py

ex-sp-ch22-07

Medium

Compare ZF and MMSE equalization for OFDM with 16-QAM over a frequency-selective channel. Plot the BER difference.

Solution
Implementation
# See code supplement ch22/python/ofdm_system.py

ex-sp-ch22-08

Medium

Implement DFT-based channel estimation: estimate HH at pilot positions, transform to time domain (IFFT), zero out taps beyond the CP length (noise suppression), and transform back (FFT). Compare MSE with linear interpolation.

Solution
Approach

DFT-based estimation exploits the channel's finite delay spread.

ex-sp-ch22-09

Medium

Simulate the effect of carrier frequency offset (CFO) on OFDM. Show how a fractional CFO of ϵ=0.1\epsilon = 0.1 (normalized to subcarrier spacing) causes inter-carrier interference (ICI).

Solution
Implementation
# Phase rotation: y[n] = x[n] * exp(j*2*pi*epsilon*n/N)
epsilon = 0.1
y_cfo = x * np.exp(1j * 2 * np.pi * epsilon * np.arange(len(x)) / N_fft)

ex-sp-ch22-10

Medium

Implement PAPR reduction using clipping-and-filtering. Show the trade-off between PAPR reduction and BER degradation.

Solution
Approach

Clip the time-domain signal to a threshold, then filter to remove out-of-band emissions. Measure both PAPR CCDF and BER.

ex-sp-ch22-11

Hard

Implement a complete OFDM radar processor: generate OFDM frames, add target returns with range and Doppler, and form the range-Doppler map. Detect targets using a 2D CFAR algorithm.

Solution
Implementation
# See code supplement ch22/python/ofdm_radar.py

ex-sp-ch22-12

Hard

Implement OFDM with adaptive modulation: assign different QAM orders to different subcarriers based on their individual SNR. Show the throughput improvement over fixed modulation.

Solution
Approach

Water-filling or SNR thresholds determine the modulation per subcarrier.

ex-sp-ch22-13

Hard

Simulate OFDMA (multi-user OFDM) where different users are assigned different subcarrier blocks. Show how frequency diversity through distributed subcarrier allocation improves cell-edge performance.

Solution
Approach

Assign subcarriers in an interleaved pattern (IFDMA) vs localized blocks (LFDMA) and compare BER for cell-edge users.

ex-sp-ch22-14

Hard

Implement SC-FDMA (DFT-spread OFDM) used in LTE uplink. Show that it has lower PAPR than OFDM while maintaining the same spectral efficiency.

Solution
Approach

SC-FDMA adds a DFT precoding step before the IFFT: the time-domain signal is a single-carrier waveform, reducing PAPR.

ex-sp-ch22-15

Challenge

Implement a complete OFDM system with LDPC coding (using a rate-1/2 LDPC code). Show the coding gain over uncoded OFDM across different channel models.

Solution
Approach

Use a sparse parity-check matrix and iterative belief propagation decoding.

ex-sp-ch22-16

Challenge

Implement MIMO-OFDM with spatial multiplexing: 2x2 MIMO with separate OFDM processing per antenna, ZF detection per subcarrier, and joint BER evaluation.

Solution
Approach

At each subcarrier, the MIMO channel is a 2x2 matrix. Apply ZF/MMSE detection independently per subcarrier.

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