**Editor’s note:** I am pleased to bring you this month’s Rarely Asked Questions (RAQ) blog by Aaron Shultz and Peter Haak, Analog Devices

**Question:**

Can you produce a frequency spectrum for all frequencies at the same time?

**Answer:**

Noise in electrical circuits is typically the enemy, and any self-respecting circuit should output as little noise as possible. Nevertheless, there are cases where a well-characterized source of noise with no other signal is entirely the desired output.

Circuit characterization is such a case. The outputs of many circuits can be characterized by sweeping the input signal across a range of frequencies and observing the response of the design. Input sweeps can be composed of discrete input frequencies or a swept sine. Extremely low frequency sine waves (below 10 Hz) are difficult to produce cleanly. A processor, DAC, and some complex, precise filtering can produce relatively clean sine waves, but for each frequency step, the system must settle, making slow work of sequential full sweeps featuring many frequencies. Testing fewer discrete frequencies can be faster, but increases the risk of skipping over critical frequencies where high Q phenomena reside.

A white noise generator is simpler and faster than a swept sine wave because it effectively produces all frequencies at the same time with the same amplitude. Imposing white noise at the input of a device under test (DUT) can quickly produce an overview of the frequency response over an entire frequency range. In this case, there is no need for expensive or complex swept sine wave generators. Simply connect the DUT output to a spectrum analyzer and watch. Using more averaging and longer acquisition times produces a more accurate output response across the frequency range of interest.

The expected response of the DUT to white noise is frequency-shaped noise. Using white noise in this fashion can quickly expose unexpected behavior such as weird frequency spurs, strange harmonics, and undesirable frequency response artifacts.

Furthermore, a white noise generator allows a careful engineer to test a tester. Lab equipment that measures frequency response should produce a flat noise profile when measuring a known flat white noise generator.

On the practical side, a white noise generator is easy to use, small enough for compact lab setups, portable for field measurements, and inexpensive. Quality signal generators with myriad settings are attractively versatile. However, versatility can hamper quick frequency response measurements. A well-designed white noise generator requires no controls, yet produces a fully predictable output.

**Noisy Discussion**

Resistor thermal noise, sometimes called Johnson noise or Nyquist noise, arises from thermal agitation of charge carriers inside a resistor. This noise is approximately white, with nearly Gaussian distribution. In electrical terms, the noise voltage density is given by

where k_{B} is the Boltzmann’s constant, T is the temperature in Kelvin, and R is the resistance. Noise voltage arises from the random movement of charges flowing through the basic resistance, a sort of R × I_{NOISE}. Table 1 shows examples at 20^{o}C.

**Table 1**

**Noise Voltage Density of Various Resistors**

A 10 MΩ resistor, then, represents a 402 nV⁄√Hz wideband voltage noise source in series with the nominal resistance. A gained up resistor-derived noise source is fairly stable as a lab test noise source, as R and T variations affect noise only by square root. For instance, a change of 6^{o}C from 20^{o}C is a change of 293 kΩ to 299 kΩ. Because noise density is directly proportional to the square root of temperature, a change of 6^{o}C temperature leads to a relatively small 1% noise density change. Similarly, with resistance, a 2% resistance change leads to a 1% noise density change.

Consider Figure 1: a 10 MΩ resistor R1 generates white, Gaussian noise at the positive terminal of an op amp. Resistors R2 and R3 gain the noise voltage to the output. Capacitor C1 filters out chopper amplifier charge glitches. Output is a 10 μV/√Hz white noise signal.

Gain (1 + R2/R3) is high, 21 V/V in this example.

Even if R2 is high (1 MΩ), the noise from R2 compared to the gained up R1 noise is inconsequential.

**Figure 1**

**Full schematic of white noise generator. Low drift micropower LTC2063 amplifies the Johnson noise of R1.**

An amplifier for the circuit must have sufficiently low input-referred voltage noise so as to let R1 dominate as the noise source. The reason: the resistor noise should dominate the overall accuracy of the circuit, not the amplifier. An amplifier for the circuit must have sufficiently low input-referred current noise to avoid (IN × R2) to approach (R1 noise × gain) for the same reason.

**How Much Amplifier Voltage Noise Is Acceptable in the White Noise Generator?**

Table 2 shows the increase in noise from adding independent sources. A change from 402 nV⁄√Hz to 502 nV⁄√Hz is only 1.9 dB in log volts, or 0.96 power dB. With op amp noise ∼ 50% of the resistor noise, a 5% uncertainty in op amp V_{NOISE} changes the output noise density by only 1%.

**Table 2**

**Contribution of Op Amp Noise**

A white noise generator could employ only an op amp without a noise-generating resistor. Such an op amp must exhibit a flat noise profile at its input. However, the noise voltage is often not accurately defined and has a large spread over production, voltage, and temperature.

Other white noise circuits may operate based on a Zener diode with far less predictable characteristics. Finding an optimal Zener diode for stable noise with µA of current can be difficult, however, particularly at low voltage (<5 V).

Some high end white noise generators are based on a long pseudorandom binary sequence (PRBS) and special filters. Using a small controller and DAC may be adequate; however, making sure that the DAC does not produce settling glitches, harmonics, or intermodulation products is something for experienced engineers. Additionally, choosing the most appropriate PRBS sequence adds complexity and uncertainty.