Pseudo-Random Number Generators in Hardware

Pseudo-random number generators, usually abbreviated as PRNGs, are circuits or algorithms that produce sequences of numbers or bits that look random, even though they are actually generated by a completely deterministic process. In hardware engineering, PRNGs are extremely useful because they can create fast, repeatable, compact, and low-cost streams of random-like data without needing a true physical noise source.

I ran into this topic in a FPGA project and I needed a simple way to understand it. This article is the result of many hours of research into the topic or PRNG and their practical use in ham radio (and not only).

This article explains what a hardware PRNG is, how it works, why engineers use it, and where it appears in practical systems. The discussion assumes average mathematical background but a solid engineering mindset. The goal is not to make the topic mysterious. In fact, one of the most important ideas is that PRNGs are usually much simpler than people first imagine.

Random versus pseudo-random

A true random source comes from a physical process whose outcome cannot be predicted exactly, at least for practical purposes. Examples include thermal noise, avalanche noise in semiconductors, radioactive decay, atmospheric noise, and jitter derived from analog physical phenomena.

A pseudo-random sequence is different. It is produced by a deterministic machine. If you know the internal state and the generating rule, you can predict every future output bit. That means a PRNG is not truly random in the physical sense. However, if the generator is designed well, the sequence can still look random enough for many engineering purposes.

This distinction matters. In engineering, “random enough for the job” is often the correct question. A simulation, a bit error rate test, a spread-spectrum modulator, or a scrambler usually does not require deep physical randomness. It requires a sequence with properties such as:

• good balance between zeros and ones,
• low predictability to casual observation,
• long repetition period,
• good autocorrelation or cross-correlation behavior,
• easy generation at high speed,
• repeatability for debugging and measurements.

Those are exactly the strengths of many hardware PRNGs.

Why hardware engineers care about PRNGs

PRNGs appear everywhere in digital and mixed-signal systems because they are cheap and useful. In hardware, especially in FPGA, ASIC, SDR, and embedded radio designs, PRNGs can generate data at line rate with very little silicon area or logic usage.

Engineers like hardware PRNGs for several reasons:

• They are simple to implement.
• They can run at very high clock rates.
• They are deterministic, so tests are repeatable.
• They can be seeded to start from known states.
• They often need only flip-flops and XOR gates.
• They are ideal for built-in self-test, simulation, coding experiments, and radio waveform generation.

In many cases, a hardware PRNG is more useful than a true random source because repeatability is a feature, not a bug. If a receiver fails only for a certain data pattern, the engineer wants to reproduce that exact pattern again.

The core idea: state machine plus update rule

At its heart, a PRNG is just a state machine.

There is an internal state, stored in registers or memory. On each clock cycle, the circuit updates that state according to a fixed rule and emits one or more output bits. The state then moves to the next value, and the process repeats.

Because the machine has a finite number of states, the sequence must eventually repeat. This is unavoidable. The only question is how long the period is and whether the repeated sequence has useful statistical properties.

A good practical way to think about it is this:

PRNG output quality depends on three things:

• the structure of the update rule,
• the size of the internal state,
• the choice of initial seed.

If the design is poor, the sequence may repeat too soon or show obvious patterns. If the design is good, it may be excellent for measurement, scrambling, spread spectrum, and simulation.

The most famous hardware PRNG: the LFSR

The most common hardware PRNG is the linear feedback shift register, or LFSR. This is the first PRNG most digital designers encounter, and for good reason: it is elegant, fast, and very hardware-friendly.

An LFSR consists of:

• a chain of flip-flops holding the current state,
• one or more taps taken from selected flip-flop outputs,
• an XOR network that combines those tap bits,
• feedback that feeds the XOR result back into the register.

On every clock edge, the register shifts, and the feedback bit enters one end of the register. The resulting stream can look noise-like, even though the logic is completely deterministic.

A small example

Suppose we have a 4-bit shift register with state bits Q3, Q2, Q1, and Q0. Imagine the feedback bit is formed by XORing Q3 and Q0. On each clock:

• the whole register shifts by one position,
• the new input bit becomes Q3 XOR Q0,
• one chosen bit, often Q0, is taken as output.

If we start from a non-zero seed such as 1001, the bits move and combine in a deterministic but complicated-looking way. Eventually the sequence repeats. With a good tap selection, the period can be maximal for that register length.

For an n-bit maximal-length LFSR, the period is:

$$ 2^n -1 $$

not \(2^n\), because the all-zero state usually gets stuck forever and is therefore excluded.

Why engineers love LFSRs

LFSRs are especially attractive because they need very little hardware. A long sequence can be generated using just flip-flops and XOR gates. In FPGA logic, this is extremely efficient. In ASIC logic, it is also compact and fast.

Another advantage is that LFSR behavior is mathematically well understood. Engineers can choose tap positions based on known primitive polynomials to obtain maximal-length sequences, often called m-sequences.

The catch: “linear” means something important

The L in LFSR stands for linear. That does not mean “easy” or “bad,” but it does mean the generator has a linear algebra structure over binary arithmetic. As a result, LFSRs are excellent for many engineering tasks but are usually not appropriate by themselves for serious cryptographic security.

In other words, an LFSR is often perfect for testing, scrambling, spreading, and measurement. It is usually not enough if your goal is to resist an adversary who is trying to predict the sequence.

Key PRNG properties engineers should understand

Period

The period is the number of output states before the sequence starts repeating. Longer periods are generally better, but period alone is not enough. A sequence can have a long period and still have poor local statistical behavior.

For example:

• a 16-bit maximal LFSR has period 65,535,
• a 23-bit maximal LFSR has period 8,388,607,
• a 31-bit maximal LFSR has period 2,147,483,647.

For many embedded systems, those are already very practical sequence lengths.

Seed

The seed is the starting state. Because the generator is deterministic, using the same seed gives the same sequence every time. That is valuable in test setups, lab measurements, and simulation environments.

For LFSRs, the all-zero state is usually forbidden because it produces no change. A non-zero seed must be loaded.

Statistical appearance

A useful PRNG should not show obvious bias or repeating short patterns. In binary form, it should usually produce nearly equal numbers of zeros and ones over a long run. It should also avoid simple visible structure when plotted or analyzed.

For many engineering uses, the sequence does not need to satisfy every possible randomness test. It only needs to behave properly for the intended application.

Correlation behavior

In communication and measurement systems, correlation often matters more than general randomness. A sequence may be attractive because its autocorrelation has a strong peak when aligned correctly and low values otherwise. This is very useful for synchronization, ranging, identification, and spread-spectrum systems.

Spectral behavior

If a PRNG drives a modulator or toggles a logic line, the resulting waveform has spectral consequences. A repeated digital pattern can generate strong discrete spectral lines. A long pseudo-random pattern can spread energy more evenly and reduce strong repetitive artifacts.

This is one reason PRNGs are useful in radio and EMC work.

Common hardware PRNG architectures

LFSR

This is the classic choice when simplicity, speed, and low hardware cost are important.

Best for:

• test pattern generation,
• CRC-like structures,
• scrambling,
• spread-spectrum sequence generation,
• BER testing,
• basic noise-like modulation experiments.

Cellular automata generators

These are less famous but still interesting in hardware. A line of cells updates according to local rules. Some rules produce complex pseudo-random behavior. They can map nicely to parallel hardware, although they are less commonly used in mainstream radio designs than LFSRs.

Counter plus nonlinear mixing

Some hardware generators start with a counter or a simple state register and apply mixing functions such as XORs, additions, rotations, or substitutions. These are more common in processors and digital systems where arithmetic logic is already available.

They can provide better statistical properties than a bare LFSR, but may cost more logic and be less transparent to analyze.

Combined generators

Two or more simpler generators can be combined to improve period or statistical behavior. For example, engineers may combine several LFSRs of different lengths or mix LFSR output with other state machines. This can reduce obvious linear artifacts while keeping the design hardware-friendly.

PRNG versus TRNG in hardware

At this point it is useful to compare PRNGs with true random number generators, or TRNGs.

Property	PRNG	TRNG
Source	Deterministic algorithm or logic	Physical noise process
Repeatability	Yes	No, or only partially
Implementation simplicity	Usually very simple	Often more complex
Speed	Usually very high	May be lower
Best use cases	Testing, scrambling, simulation, spread spectrum	Key generation, entropy source, security seeding

A good practical engineering pattern is this: use a TRNG to obtain entropy for seeding, then use a PRNG to expand that entropy into a fast stream. That is common in secure digital systems. But in many radio or laboratory uses, an ordinary PRNG alone is entirely sufficient.

Why pseudo-random sequences matter in radio engineering

Radio systems often need signals that are broadband, decorrelated, repeatable, and easy to synchronize against. PRNG-derived sequences fit these needs extremely well.

There are at least six important uses in radio engineering.

Scrambling to avoid long repetitive patterns

If a data stream contains long runs of zeros or ones, the transmitted waveform can develop undesirable structure. Clock recovery can suffer, spectral lines may become more pronounced, and measurements may become less representative.

A scrambler uses a PRNG-like sequence, often implemented with an LFSR, to randomize the transmitted bit stream. The receiver applies the corresponding descrambler to recover the original data.

This does not necessarily provide security. Its main purpose is signal conditioning: producing a data stream that is more suitable for transmission and processing.

Spread-spectrum systems

One of the most famous radio uses of PRNG sequences is direct-sequence spread spectrum. Here, the information signal is multiplied or XORed with a much faster pseudo-random chip sequence. This spreads the energy across a wider bandwidth.

The advantages can include:

• resistance to narrowband interference,
• processing gain,
• multiple-access capability when different users have different codes,
• noise-like appearance of the transmitted signal.

In such systems, the choice of sequence and its correlation properties are critical. LFSR-based m-sequences and related families are historically very important here.

Frame synchronization and correlation detection

Known pseudo-random sequences are useful as preambles, sync words, or training sequences. Because the receiver knows the exact expected pattern, it can correlate the incoming signal with the local copy and detect timing or alignment.

This is extremely valuable in digital receivers, SDR experiments, and packet radio.

Bit error rate testing

When engineers test a radio link, they often do not want to transmit plain counting patterns such as 0000, 0001, 0010, and so on. Those patterns are too artificial and may not stress the system in a realistic way.

Instead, a PRNG generates a test sequence, typically called a PRBS, meaning pseudo-random binary sequence. The receiver compares the received sequence against the expected local copy and counts errors. This gives bit error rate, burst error behavior, synchronization margins, and performance under noise or fading.

Noise-like beaconing and ranging experiments

Because a pseudo-random sequence can have a sharp correlation peak, it can be used for ranging, delay estimation, and channel sounding. A transmitter emits a coded sequence, and the receiver performs correlation to estimate arrival time or channel impulse response.

That idea appears in many forms, from advanced radio systems to modest SDR-based experiments.

EMI and spectral spreading

Periodic digital clocks and repetitive bit patterns can produce strong narrow spectral components. In some systems, controlled pseudo-random modulation or dither helps spread energy and reduce concentrated peaks. This can improve spectral behavior in converters, clocks, and digital transmit chains.

The exact usefulness depends on regulations and design details, but the engineering principle is important: pseudo-random activity can make a system less obviously periodic.

PRNG examples in ham radio and amateur experimentation

This is where the topic becomes especially fun for radio hobbyists. Amateur radio operators, SDR experimenters, and homebrew builders can use PRNGs in many practical ways without needing exotic hardware.

Here are several realistic examples.

Building a PRBS test source for an SDR link

Suppose a hobbyist is developing a digital UHF link using two SDRs or a small FPGA modem. Instead of sending text or audio, the experimenter can transmit a PRBS pattern generated by an LFSR. At the receiver, the same PRBS is generated locally and compared against the received bits.

This allows the operator to measure:

• bit error rate,
• sensitivity to antenna alignment,
• performance versus signal-to-noise ratio,
• the effect of filters, symbol timing, and equalization.

This is far more informative than simply checking whether a payload “seems to work.”

Creating a scrambler for a homebrew digital modem

In packet-like digital systems, repetitive content can create bad spectral behavior or weak timing transitions. A ham experimenting with custom modulation can use an LFSR scrambler before modulation and a matching descrambler after demodulation. This makes the transmitted signal behave more like real traffic and often improves receiver operation.

Spread-spectrum experimentation in the amateur lab

Where regulations and band plans permit experimentation, pseudo-random chip sequences can be used to study direct-sequence spread-spectrum techniques. Even if the final on-air use is restricted, the bench experiment itself is valuable. The hobbyist can observe:

• how spreading widens occupied bandwidth,
• how despreading recovers the wanted signal,
• how narrowband interference is suppressed after correlation,
• how code synchronization becomes the central design challenge.

This kind of experiment is excellent for SDR education because it connects digital logic, RF bandwidth, and signal processing in one project.

Receiver testing with repeatable noise-like input

A hobbyist can use a PRNG-driven baseband source to create realistic symbol streams or pseudo-noise modulation for receiver stress testing. Because the pattern repeats only after a long period, the signal behaves more like random traffic than a simple test tone or square wave.

This can help evaluate AGC response, clock recovery, burst handling, and decoder robustness.

Antenna and propagation experiments using coded signals

An interesting amateur project is coded beaconing. Instead of transmitting only an unmodulated carrier or a simple tone, the experimenter can transmit a known pseudo-random sequence. A remote SDR or receiver can correlate against that sequence and pull weak signals out of noise more effectively than with simple threshold detection.

This idea is conceptually related to matched filtering and coherent detection. It can be very educational for weak-signal work.

Measuring channel delay or multipath in SDR projects

A homebrew station using GNU Radio, FPGA logic, or microcontroller-assisted DSP can transmit a pseudo-random sequence and examine the correlation output at the receiver. Peaks in the correlation function can reveal direct and reflected paths. That is a practical way to study multipath indoors, around buildings, or across local terrain.

For an amateur experimenter, this is a beautiful example of how a simple digital sequence turns into a real propagation measurement tool.

A practical engineer’s view of LFSR-based PRBS patterns

In test equipment and digital communications, you will often see names such as PRBS7, PRBS9, PRBS15, PRBS23, or PRBS31. These names usually indicate an LFSR-generated pseudo-random binary sequence with a period tied to the register length.

For example:

• PRBS7 has period \(2^7-1\) = 127
• PRBS15 has period \(2^{15} – 1\) = 32,767
• PRBS23 has period \(2^{23} – 1\) = 8,388,607
• PRBS31 has period \(2^{31} – 1\) = 2,147,483,647

Shorter PRBS patterns are easier to debug and observe. Longer PRBS patterns better approximate random traffic and are more likely to expose rare weaknesses in a link.

A good engineering practice is to use more than one pattern length during validation. A receiver that passes PRBS7 may still fail under PRBS23 or PRBS31 because longer histories expose different corner cases.

What a hardware implementation looks like

From a circuit point of view, a simple hardware PRNG can be astonishingly small.

An LFSR implementation typically contains:

• a register of n flip-flops,
• XOR gates for selected taps,
• reset and seed loading logic,
• optional parallel output logic,
• optional enable and synchronization control.

In FPGA design, this is often just a few lines of HDL. In ASIC design, it is similarly compact. In discrete logic, it can be built from standard shift registers and XOR gates for educational projects.

Serial versus parallel generation

A basic PRNG emits one bit per clock. But many systems need multiple bits per cycle. In that case, the generator can be unrolled or advanced multiple steps per clock. This is common in high-throughput digital systems.

Seeding and reset behavior

Every practical design needs a clean way to load a seed. Otherwise the initial state may be undefined, which is dangerous in test systems. For repeatable results, engineers often document the exact seed value used in a measurement.

Synchronization at the receiver

When the same pseudo-random sequence is needed at both ends of a link, the receiver must know the sequence phase. This can be handled by:

• sending a known sync word,
• resetting both generators at known times,
• using correlation to find alignment,
• embedding the PRNG in a self-synchronizing scrambler structure.

In practice, synchronization is often harder than sequence generation itself.

Limitations and common misunderstandings

Pseudo-random does not mean secure

This is the biggest misunderstanding. Many hardware PRNGs, especially plain LFSRs, are not cryptographically secure. They are deterministic and often mathematically reconstructible from observed outputs.

That does not make them useless. It just means they should be used for the right jobs.

Long period does not guarantee quality

A very long period sounds impressive, but it is only one metric. Correlation behavior, bias, implementation correctness, and application-specific properties may matter more.

A good PRNG can still be wrong for the system

A sequence that is excellent for BER testing may be poor for cryptography. A sequence that is fine for scrambling may not be ideal for multi-user code division experiments. “Good” always means “good for a specific purpose.”

Hardware bugs can ruin the intended properties

An incorrect tap choice, a broken reset, a forbidden zero seed, or a synthesis mistake can completely change the sequence. Engineers should always verify the actual generated pattern, period, and correlation behavior rather than assuming the HDL is correct.

A simple mental model for non-specialists

For readers who do not want the formal math, here is a useful mental model.

Imagine a row of memory bits. Every clock cycle, the bits shift, and a new bit is created by combining a few of the old bits through XOR. That one simple rule causes the register to pass through a long chain of states that looks irregular. Because the rule is deterministic, the sequence is reproducible. Because the state transitions are chosen carefully, the output looks random enough for many engineering tasks.

That is the essence of a hardware PRNG.

Where the math really matters, and where it does not

You do not need advanced mathematics to use PRNGs effectively in engineering. For many practical tasks, you only need to understand:

• the generator has internal state,
• the state updates once per clock,
• the sequence eventually repeats,
• some structures produce much better sequences than others,
• LFSRs are common because they are simple and efficient.

More advanced math becomes useful when you start selecting primitive polynomials, proving maximal length, analyzing linear complexity, or studying correlation families for spread-spectrum systems. Those are important topics, but they are not required to begin using PRNGs intelligently.

Practical design advice for engineers and hobbyists

• Start with an LFSR unless you have a strong reason not to.
• Use a known-good tap set for the chosen register length.
• Never seed a standard LFSR with all zeros.
• Document the seed used in tests.
• Check period and output statistics in simulation before hardware deployment.
• For radio work, pay attention to correlation and spectral behavior, not just period.
• Do not treat a plain LFSR as a secure cryptographic generator.
• When debugging links, use repeatability to your advantage.

Conclusion

Pseudo-random number generators in hardware are one of those engineering ideas that are both simple and powerful. They are deterministic machines that produce random-like sequences, and that combination turns out to be extremely useful. Hardware PRNGs are compact, fast, reproducible, and ideal for testing, scrambling, synchronization, spread-spectrum work, and many radio experiments.

For radio science and ham radio, PRNGs are especially valuable because they connect digital logic with real RF behavior. A simple LFSR can become a scrambler, a PRBS test source, a synchronization pattern, a channel sounding tool, or the basis of a spread-spectrum experiment. That makes PRNGs more than a digital design curiosity. They are practical instruments for understanding and building communication systems.

If there is one idea to remember, it is this: a hardware PRNG is not “fake randomness” in a dismissive sense. It is a controlled source of random-like structure, and in engineering, control is often exactly what makes it so useful.

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