The figure is today’s second flowgraph, L20-ofdm-link.grc, running. It sends a line of text as packets, and each packet travels as OFDM: 48 narrow subcarriers side by side, the idea Lesson 14 found beside an HD Radio station. Each subcarrier carries 16-QAM, a constellation of 16 points on a grid, so every symbol on every subcarrier carries 4 bits. The spectrum is the flat block of all 52 subcarriers, 48 for data and 4 pilots, the left constellation shows the subcarrier points as the receiver’s FFT delivers them, and the right one shows them after the receiver’s equalizer has corrected each subcarrier. The left points lie on rings because the receiver starts its FFT a few samples early, inside the guard, which turns each subcarrier by its own angle; the equalizer measures that turn and removes it.

Today we build both halves. We start with QAM on a single carrier, following GNU Radio’s constellation objects and Constellation Modulator block and PySDR’s Digital Modulation chapter, and we watch noise act on QPSK, 16-QAM, and 64-QAM. Then we follow the OFDM transmitter and receiver that ship with GNU Radio as the examples tx_ofdm.grc and rx_ofdm.grc, with the blocks of the GNU Radio wiki’s Basic OFDM Tutorial, and we put Lesson 15’s multipath channel and a longer echo in front of them. We finish with which systems use each one and how to recognize it on a live band.

Lesson 15’s PSK put every point on one circle, so only the phase carried data. Quadrature amplitude modulation (QAM) also uses the amplitude: the points fill a square grid, and each point is one level of \(I\) and one level of \(Q\). 16-QAM has 4 levels on each axis and 64-QAM has 8, so their symbols carry \(\log_2 16 = 4\) and \(\log_2 64 = 6\) bits. QPSK is the smallest grid, 2 by 2. Lesson 3 drew decision regions for PAM and PSK; on a QAM grid each point’s region is the square around it.

The figure is L20-qam-noise.grc with almost no noise. Each modulation sends 8,000 symbols per second through the same RRC filter as Lesson 15, with \(\alpha = 0.35\), so the spectrum below shows all three filling the same 10.8 kHz. At the same symbol rate, 16-QAM carries twice the bits of QPSK and 64-QAM three times as many.

A Constellation Object block holds a constellation as a variable. GNU Radio has 16-QAM built in; the QPSK and 64-QAM objects list their points, \(\pm 1 \pm 1j\) and the 8 by 8 grid of odd numbers from \(-7\) to \(7\), and scale them to an average power of 1, as the built-in 16-QAM is. The Constellation Modulator takes bytes, splits them into groups of bits, maps each group to a point, and shapes the symbols with the RRC filter, 4 samples per symbol here. This is the block Lesson 15’s multipath flowgraph used for QPSK.

Each row then adds noise in a Channel Model and runs the receiver of Lesson 15: the matched filter and Symbol Sync. All three Channel Models take their noise amplitude from the same slider, so every constellation sees the same noise. The flowgraph has no frequency offset, so it needs no carrier recovery.

The figure raises the noise amplitude from 0.05 to 0.3. The noise is the same in all three channels, and it spreads every point into a cloud of the same size. QPSK’s four points sit far apart, and its clouds stay well separated. 16-QAM’s clouds spread toward each other, and a stray point crosses into a neighbor’s region now and then. 64-QAM’s points sit closest together, and its clouds have merged into one square with no gaps, so the receiver picks wrong points constantly.

Every QAM link faces this tradeoff. More points carry more bits in the same band, but they sit closer together, so they need a cleaner signal. PySDR’s Digital Modulation chapter runs the same comparison with BPSK, QPSK, and 16-QAM.

A PSK receiver ignores the amplitude, because every point sits on the same circle, but a QAM receiver cannot. The Receiver gain slider in L20-qam-noise.grc scales the received points, as a receiver with the wrong level would, and the figure sets it to 0.6. QPSK’s points shrink but stay in their quadrants, so a QPSK decoder still reads every one. 16-QAM’s outer points move from \(\pm 0.95\) to about \(\pm 0.57\), inside the lines at \(\pm 0.63\) that separate outer points from inner ones, so a decoder that expects the full-size grid reads every outer point as an inner one.

A real QAM receiver therefore runs an AGC to hold the average power where the constellation expects it, as the L15 Lab’s PlutoSDR flowgraph did, and its amplifiers have to stay out of compression, which would squeeze the outer points inward. Carrier recovery for QAM uses GNU Radio’s Constellation Receiver block in place of the Costas Loop, and a final project that sends QAM over a radio would start there.

Lesson 14 met OFDM in HD Radio’s sidebands. Orthogonal frequency-division multiplexing divides a channel among many subcarriers spaced so that each one is zero at the centers of all the others, which lets the receiver separate them with an FFT. Each OFDM symbol is a list of constellation points, one per subcarrier: the OFDM Carrier Allocator places the points, an inverse FFT turns the list into time samples, and the OFDM Cyclic Prefixer adds the guard. HD Radio puts QPSK on its subcarriers; today’s link puts 16-QAM on them.

Many slow subcarriers have one large advantage over a single fast carrier. Each OFDM symbol lasts much longer than one symbol of a single carrier with the same total bit rate, so an echo a few samples late smears only a small part of it, and the guard absorbs that part. The next frames show this with Lesson 15’s multipath taps.

The top row is the transmitter of GNU Radio’s tx_ofdm.grc example. Each packet gets a header in BPSK and a payload in 16-QAM; the Carrier Allocator places them on 48 data subcarriers with 4 pilot subcarriers of known value, and puts two known preamble symbols at the start of every packet. The FFT is 64 points and the guard is 16 samples, at 100,000 samples per second, so the subcarriers sit 1,562.5 Hz apart. The middle row is the channel: a Channel Model with Lesson 15’s four tap sliders, an echo, and noise.

The bottom rows are the receiver of rx_ofdm.grc. The Schmidl & Cox OFDM synch block finds the start of each packet from the first preamble symbol. The FFT of the second preamble symbol, whose values the receiver knows, gives the channel’s gain and phase on every subcarrier, which is the channel estimate. The OFDM Frame Equalizer divides each subcarrier by its gain and turns it back by its phase, and the decoders read the header and then the 16-QAM payload.

The guard interval, also called the cyclic prefix, is a copy of the last 16 samples of each symbol placed in front of it. The figure shows why it is there. An echo arrives 8 samples after the direct signal, so the end of its symbol 1 overlaps the start of the direct path’s guard for symbol 2. The receiver’s FFT window skips the guard (in this receiver, about 9 of its 16 samples; see two frames ahead), so it sees only symbol 2 from both paths: the direct one, and the echo’s copy, which is the same symbol turned and scaled. On each subcarrier that sum is one point with a new gain and phase, which the equalizer removes.

An echo later than the guard reaches past it into the FFT window, and part of symbol 1 lands in symbol 2’s FFT, where the equalizer cannot remove it. In the link flowgraph the guard lasts 16 samples, or 160 µs, and Lesson 14’s explorer showed the same effect with HD Radio’s guard.

The four tap sliders use the expression of L15-psk-multipath.grc: each sets the channel’s gain at one frequency. At this flowgraph’s 100,000 samples per second those frequencies move outward, so Tap 1 now sets the gain near the lower edge, at \(-40\) kHz, and Taps 2, 3, and 4 sit at \(-20\), 0, and \(+20\) kHz, all inside the OFDM block. At Lesson 15’s settings the spectrum tilts, weak at the lower edge and strong at the upper one. Before the equalizer the subcarrier points are a jumble, because every subcarrier has a different gain and phase. After it, they form the 16-QAM grid again, and the text arrives.

Lesson 15 needed an adaptive equalizer, a filter that learned the inverse of the channel, to undo the same tap settings on a single QPSK carrier. OFDM needs only one complex number per subcarrier, measured from the preamble. The clusters for the weakest subcarriers come out a little larger, because the equalizer amplifies their noise along with their signal; with the noise amplitude at 0.05 the weak edge corrupts about half the lines.

The Echo delay and Echo level sliders add a second copy of the signal, delayed and weaker, like a reflection from a building. An echo 8 samples late at half the amplitude makes the spectrum ripple: the two copies add at some frequencies and partly cancel at others, eight times across the 100 kHz window. Each subcarrier lands at its own place on that ripple, so before the equalizer the points fall on many separate arcs.

The echo ends inside the 16-sample guard, so each subcarrier sees only a fixed change in gain and phase. The channel estimate measures it, the equalizer removes it, and every line arrives. A single-carrier link carrying the same bit rate in the same band would see each symbol smeared across its neighbors, as Lesson 15’s clusters were.

At 20 samples the echo is longer than the guard, and the end of each symbol’s echo reaches into the next symbol’s FFT window. That leakage acts like noise that the equalizer cannot remove, and the grid falls apart even though the noise amplitude has not changed.

The text breaks before the echo reaches the full 16 samples. The receiver starts its FFT window about 7 samples before the end of the guard, as a margin for its own timing error, which leaves about 9 samples of the guard for echoes. Step the echo delay up one sample at a time from 8 and watch where the lines start to fail. A real system chooses its guard to be longer than the echoes it expects, and the next frame lists some of those choices.

OFDM serves wireless systems that meet echoes: Wi-Fi indoors, LTE and 5G across a city, and digital TV and HD Radio across a whole region, where a guard of hundreds of microseconds covers reflections from kilometers away. A 20 MHz 802.11a or 802.11g Wi-Fi channel uses exactly the subcarrier layout of today’s flowgraph, 48 data and 4 pilot subcarriers in a 64-point FFT, at 20 million samples per second, so its subcarriers are 312.5 kHz apart. LTE uses 15 kHz, and 5G NR uses 15 kHz or a wider multiple of it. DVB-T is Europe’s digital TV; the newer US standard, ATSC 3.0 or NextGen TV, is OFDM too.

Single-carrier QAM serves channels with few echoes, such as cable TV and cable modems on coaxial cable, which carry 64-QAM and 256-QAM in 6 MHz channels (DOCSIS 3.1 moved cable modems to OFDM as well), and point-to-point microwave links between towers. Coaxial cable keeps the signal off the air, so an SDR antenna cannot receive it.

The two choices answer different questions. OFDM or a single carrier depends on the channel’s echoes. The constellation depends on the noise: in the figure the noise amplitude is 0.15, the 16-QAM clouds touch, and most lines arrive with errors, where QPSK on the same subcarriers would still deliver them.

Wi-Fi and LTE make the second choice continuously. A receiver close to the access point or tower, with a clean signal, gets 64-QAM or more on every subcarrier; as it moves away and the signal weakens, the link steps down to 16-QAM, then QPSK, and adds more error-correcting code. PySDR’s Channel Coding chapter calls these combinations modulation and coding schemes. HD Radio, which must reach car radios at the edge of a station’s coverage, stays with QPSK.

These signals are on the air around you, and the PlutoSDR shows their spectrum. We look only at the spectrum; we do not decode anyone’s traffic. L20-band-pluto.grc is a PlutoSDR Source with a frequency sink and a waterfall, a Center frequency control you change while it runs, and a Sample rate choice of 2 MS/s or 20 MS/s. It opens at 2 MS/s and 100.1 MHz. The course Plutos run from 70 MHz to 6 GHz, so all three signals are in range.

A Pluto streams about 4 MS/s continuously over USB 2.0, and a Wi-Fi channel is 20 MHz wide. At 20 MS/s the Pluto still fills each buffer of 131,072 samples without a break, 6.5 ms of signal, but the laptop cannot take the buffers as fast as they come, so samples are lost between them. The spectrum needs only a short unbroken stretch, so the displays still show the channel’s shape; they skip some of what happens between buffers.

1. At 2 MS/s, find an FM station near 100 MHz that carries HD Radio, and set the center frequency about 400 kHz to one side of it, so the Pluto’s own mark at 0 Hz stays off the station and both of Lesson 14’s flat shelves. If the whole spectrum rises and spreads, lower the RX gain.

2. Choose 20 MS/s and tune to 2.437 GHz, Wi-Fi channel 6, and watch the waterfall. Short bursts come and go. Those carrying data are OFDM: a flat top with steep edges about 17 MHz wide, often with a small dip at the center, where the subcarrier at 0 Hz is left empty. The Pluto marks its own 0 Hz too, so tune 2 MHz higher: the dip that moves with the channel is Wi-Fi’s. Some access points still send beacons in the older 802.11b format, which look rounded instead.

3. Still at 20 MS/s, tune between 729 and 756 MHz, where US carriers run LTE downlinks; which frequencies are in use varies by area. A 10 MHz LTE carrier shows as a flat top about 9 MHz wide with steep edges, and unlike Wi-Fi’s bursts it stays on.

4. At 2 MS/s, find the same LTE carrier. The window shows only a flat slice of it, so step the center frequency 2 MHz at a time to find both edges.

As in every lesson, receiving is unrestricted and transmitting is not; nothing in this lesson transmits.

L20 is the last lesson of the course. The rest of Module 7 is the final project, in which you choose a signal to build, receive, or decode, write the success criteria we will grade it against, and demonstrate it working. Every lesson’s flowgraphs are starting points, and today’s two are a direct route into a project on a modern waveform: send a file over L20-ofdm-link.grc through a PlutoSDR and a cable with an inline 30 dB attenuator and an AGC, add a 64-QAM payload by copying the qam64 Constellation Object from L20-qam-noise.grc into payload_mod, or measure the spectrum of the Wi-Fi and LTE bands around the Academy.