Physics of Wireless Transmission

The Switching and Routing course counted seventeen separate disagreements, and in every one of them accepted a single thing without asking: if a cable connects two devices, the frame leaving that cable reaches the other end. The link either exists or it does not. No third state besides breakage was ever considered; if the tables disagree the packet goes to the wrong place, but on its way to where it should go, it is never lost.

In wireless this acceptance falls away. The medium belongs to no one, its boundary is not drawn, and transmission is not a certainty but a probability. A client’s link is not “present” or “absent”; it is a curve — it falls with distance, drops with interference, descends in tiers, and reaches zero somewhere. This course’s question is born right here: what is the gap between the boundary the design intends and what actually gets through? This lesson builds the wireless half of that gap and measures its most counterintuitive result: being inside the cell is not the same as working.

The Shared Medium

In a wired link the transmission medium is split between two ends: whichever side plugs in the cable uses it. In wireless the medium is shared and there is no record of who is using it. Every transmitter broadcasting in the same frequency range is noise to every receiver listening on that range — it makes no difference whether the transmitter belongs to the same network.

For this reason the usable frequency range is split into a band, and the band is divided into channels. Two neighbors broadcasting on the same channel directly corrupt each other; two broadcasts on adjacent channels can also bleed into each other from spillover. A channel plan’s job is to make sure that two access points close enough to hear each other do not land on the same channel.

The table below is the fictional band layout this lesson uses. Real frequency allocations vary from country to country, are updated by regulatory decisions, and affect nothing this lesson measures; the numbers here are chosen only to show the shape of channel planning.

# taught band layout , fictional and not executed

band N (narrow)  : 8 channels, channel width 1 unit
                   non-overlapping channel set: 1, 4, 7  (three channels)
                   range coefficient high, tier ceiling low

band W (wide)    : 24 channels, channel width 1 unit
                   non-overlapping channel set: all 24 channels
                   range coefficient low, tier ceiling high

bonding          : four channels in band W can bond into one 4-unit channel
                   -> 6 non-overlapping wide channels, tier ceiling doubled

The narrow band has only three non-overlapping channels; a fourth access point inevitably reuses a neighbor’s channel. The wide band has channels to spare, but range is short. This trade-off will be measured in later lessons of the course; here only where interference comes from is being established.

The effect of channel and band width on the transport layer was established in the Network Models course; it is not repeated here. That a shared medium produces collisions is not new either — the Switching and Routing course’s first lesson counted this on the hub. Where this lesson departs is the measure itself: there, what was measured was the number of colliding frames; here, what is measured is in how many clients, and in which two directions, the design’s promise diverges.

Signal Strength

What the receiver sees is called signal strength, and two things bring it down: distance and interference. This lesson combines the two on a single fictional scale.

$$\text{signal} = \text{power} - \text{distance} - \text{interference}$$

Here power is the access point’s signal budget, distance is in meters, and interference is the contribution from neighbors using the same channel at that client’s location. All three are fictional units.

In real propagation, attenuation is not a linear function of distance; walls, reflections, and bodies each take their own share. The linear form here is a fiction chosen to keep the measurement readable, and all of this lesson’s claims live inside that fiction. The one thing the fiction carries is this: signal strength is a continuous quantity, and it drops for two independent reasons.

Rate Tiers

Even though signal strength is continuous, transmission is not. Based on the signal strength it measures, the transmitter picks a rate tier: a dense modulation when the signal is high, a more resilient but slower modulation as it drops. Tier transitions are threshold-bound.

Signal strength	Rate tier	Meaning
34 and above	6	full rate
28 – 33	4	one tier down
22 – 27	2	two tiers down
18 – 21	1	lowest tier
below 18	0	cannot associate

The last row is a separate threshold. When signal strength drops below 18, even the most resilient modulation does not hold, and the client cannot associate. Association is the establishment of a link between the client and the access point; until it is established, there is no network for the client.

The existence of tiers is what separates wireless from a wired link. A wired link either works or breaks; a wireless link can run six times slower, and no connection status indicator reports this. Indicators show association, not tier.

The Asymmetric Bond and Shared Time

The calculation above was made for a single direction. In reality every link has two directions, and the two are not equal. The access point is a mains-powered device with a high signal budget; the client runs on a battery and its budget is low. The result is an asymmetric link: the client may hear the access point while the access point does not hear the client. On the user’s side the network is visible, sits in the list, can be selected — and association never forms. Because this lesson measures a single direction, it does not put a number on the asymmetry; but this is the source of the “I can see the network but cannot connect” symptom, and it does not look like a coverage problem.

The second difference is that the medium is half duplex. While a transmitter is broadcasting, its own transmission fills its own receiver; it cannot hear anyone else broadcasting at the same time. In a wired link, collision is sensed — the sender listens to the signal and sees the corruption. In wireless it cannot be sensed, so the mechanism is built on avoiding collision rather than detecting it: the transmitter listens to the medium, and if it finds it idle, waits a further random interval before starting to transmit. There is no referee for who uses the medium; order is born from the waiting itself.

When these two facts combine, the measured effect of a tier drop multiplies. A client at tier 1 holds the medium six times as long as a client at tier 6 to carry the same data; this follows directly from the definition of the tier numbers. Because the medium is shared, everyone in the cell waits for that entire span. A single slow client at the cell’s edge slows down every fast client at the center. In the measurement, clients that drop to lower tiers do not just lose their own share; they diminish everyone else’s too.

The Design’s Promise

The promise made when a wireless network is designed is a radius: “the client inside the thirty-meter cell runs at full rate.” This promise is the course’s intent, and it is the measurement’s oracle. Because we write the fiction ourselves, we know every client’s real distance; what the intent means is beyond dispute.

The medium’s answer, on the other hand, is a separate function: can the client associate? Comparing the two produces four outcomes, and the whole course counts this foursome.

• Correct admit — intent would admit, the medium admitted.
• Correct deny — intent would deny, the medium denied.
• False admit — intent would deny, the medium admitted. A client outside the cell associated.
• False deny — intent would admit, the medium denied. A client inside the cell could not associate.

The two directions are counted separately. A measurement that counts only false admit cannot see where coverage falls short; one that counts only false deny cannot see leaking coverage. Throughout the course both stand side by side in every table.

The measurement’s assumptions:

• WN1. Forty clients’ distance and the interference they see come from a single generator. Distance is in the 5–45 m range, interference in 0–12 units; the oracle is known because we produced the fiction ourselves.
• WN2. Signal strength is linear, in the form power − distance − interference. This is not a propagation model; it is a fiction that keeps the tiers readable.
• WN3. There is a single access point, and its signal budget is 52. This lesson does not set up another power value; a power sweep is the subject of the channel planning lesson.
• WN4. The cell radius is 30 m, and intent looks only at distance. Intent does not see interference — the design does not know what neighbors will do when it makes its promise.
• WN5. The tier thresholds are 34, 28, 22, and 18; 18 is also the association threshold. The medium’s decision is read as rate_tier > 0, which is association itself.
• WN6. The set is forty clients; the smallest measurable difference in this set is 1/40 = 0.025. A difference smaller than this is not defended by this measurement.

Measurement

"""Physics of wireless transmission: signal strength, rate tier, and the cell's promise."""
SEED = 20260811
POWER = 52           # access point's signal budget, fictional unit
RADIUS = 30          # cell radius the design promises, meters
TIERS = ((34, 6), (28, 4), (22, 2), (18, 1))   # (signal threshold, rate tier)


def generator(seed):
    d = seed % 2147483646 + 1

    def r(n):
        nonlocal d
        d = (d * 48271) % 2147483647
        return d % n
    return r


def clients(count=40, seed=SEED):
    r, pool = generator(seed), []
    for i in range(count):
        pool.append({"no": i + 1, "distance": 5 + r(41), "interference": r(13)})
    return pool


def signal(c, power=POWER):
    return power - c["distance"] - c["interference"]


def rate_tier(c, power=POWER):
    for threshold, tier in TIERS:
        if signal(c, power) >= threshold:
            return tier
    return 0


def cell_intent(c, radius=RADIUS):
    """The design's promise: the client inside the cell runs at full rate."""
    return c["distance"] <= radius


def medium(c, power=POWER):
    return rate_tier(c, power) > 0


def gap(events, intent, mechanism):
    d = {"correct_admit": 0, "correct_deny": 0, "false_admit": 0, "false_deny": 0}
    for e in events:
        n, m = intent(e), mechanism(e)
        if n and m:
            d["correct_admit"] += 1
        elif not n and not m:
            d["correct_deny"] += 1
        elif m:
            d["false_admit"] += 1
        else:
            d["false_deny"] += 1
    return d


pool = clients()
print(f"clients {len(pool)} | distance {min(c['distance'] for c in pool)}-"
      f"{max(c['distance'] for c in pool)} m | interference "
      f"{min(c['interference'] for c in pool)}-{max(c['interference'] for c in pool)} | "
      f"inside cell {sum(1 for c in pool if cell_intent(c))}")
print()
print(f"{'tier':>6s} {'clients':>8s} {'lowest signal':>14s}")
for threshold, tier in TIERS:
    k = [c for c in pool if rate_tier(c) == tier]
    print(f"{tier:6d} {len(k):8d} {min((signal(c) for c in k), default=0):14d}")
zero = [c for c in pool if rate_tier(c) == 0]
print(f"{0:6d} {len(zero):8d} {max(signal(c) for c in zero):14d}")
print()

d = gap(pool, cell_intent, medium)
print(f"{'correct admit':>13s} {'correct deny':>12s} {'false admit':>12s} "
      f"{'false deny':>12s}")
print(f"{d['correct_admit']:13d} {d['correct_deny']:12d} {d['false_admit']:12d} "
      f"{d['false_deny']:12d}")
print()

inside = [c for c in pool if cell_intent(c)]
print(f"{'tier inside cell':>17s} {'clients':>8s}")
for tier in (6, 4, 2, 1, 0):
    print(f"{tier:17d} {sum(1 for c in inside if rate_tier(c) == tier):8d}")
print()
print("false deny (no, distance, interference, signal):",
      [(c["no"], c["distance"], c["interference"], signal(c)) for c in inside if rate_tier(c) == 0])

clients 40 | distance 6-44 m | interference 0-12 | inside cell 27

  tier  clients  lowest signal
     6        9             35
     4        4             30
     2        8             22
     1        2             18
     0       17             17

correct admit correct deny  false admit   false deny
           23           13            0            4

 tier inside cell  clients
                6        9
                4        4
                2        8
                1        2
                0        4

false deny (no, distance, interference, signal): [(4, 24, 11, 17), (12, 30, 6, 16), (25, 30, 12, 10), (32, 30, 7, 15)]

Associating Is Not the Same as Working

The second table distributes forty clients across tiers. 9 clients are at full rate, 4 are one tier down, 8 are two tiers down, 2 are at the lowest tier, and 17 cannot associate at all. The total associated is 23.

The third table compares intent with the medium. There are 27 clients inside the cell; 23 of them associate, 4 cannot. None of the 13 clients outside the cell associate: false admit 0. At this power the boundary does not leak. An audit that looked only at this column would declare the configuration flawless.

The fourth table gives what that audit cannot see. The tier distribution of the 27 clients inside the cell is 9, 4, 8, 2, 4. The design’s promise was “the client inside the cell runs at full rate.” The number of clients for whom that promise is truly kept is 9 — a third of those inside the cell. 14 clients associate, but at lower tiers, and 4 cannot associate at all.

The reading that follows is the course’s fourth reading: the boundary is not binary. The distinction between “admitted” and “not admitted” is a coarse reading of the tier sequence. A configuration can show zero error on this coarse reading while failing to keep two-thirds of the promise it made.

Where the four false-deny clients sit is instructive too. All four are inside the cell but close to its edge: their distances are between 24 and 30 m, and the interference they see is high. Distance alone is not enough — intent never sees interference at all (WN4). When the design draws the cell, it does not know which channel the neighboring access points will broadcast on; interference is a quantity that appears after the promise has already been made.

This shows why counting the two directions separately is necessary. Leakage is zero, the shortfall is four. A report that wants a single number will call this configuration either flawless or flawed; either way it will be wrong.

Four clients’ share of a forty-client set is 0.100. Since the smallest measurable difference in this set is 1/40 = 0.025 (WN6), four sits comfortably inside the measurement band and cannot be called noise. A one-client difference, by contrast, is defensible with this set but weak; this bound will be recalled throughout the course whenever single-client shifts are interpreted.

Summary

• In wireless the medium is shared and transmission is not a certainty; the link is not a binary state but a curve that falls with distance and drops with interference.
• Signal strength strikes tier thresholds at 34, 28, 22, and 18. The last threshold is also the association threshold; there is no network for a client that drops below it.
• In the forty-client fiction at power 52 the tier distribution is 9, 4, 8, 2 and 17 unable to associate; the total associated is 23.
• In the same configuration false admit is 0 and false deny is 4: the boundary does not leak, but it leaves four clients inside the cell out.
• Only 9 of the 27 clients inside the cell are at the full rate tier. Associating is not the same as working, and a one-directional audit cannot see it.

Next Step

This measurement counts the decision a single access point makes on its own: it associates the client if it hears it, and does not if it does not. In a building, though, there are dozens of access points, and each one’s signal budget, channel, and threshold is a separately given value. The next lesson measures where that decision is made: does each access point decide by its own view, or does a shared controller decide? If the same client can get two different answers from two access points, the owner of the answer is not the client’s condition but which one it happens to land on.