# Gaussian signal generation circuit

I was studying op amp application and saw the square wave, triangular wave, sine wave. I was wondering if it is possible to generate Gaussian shape wave with op amp. I am basically trying to modulate sine wave signal and Gaussian wave signal and send it to a short range pcb antenna. I am a bit newbie in signal processing so pardon me if I ask any silly question.

If Gaussian wave generation is not possible with op amp, Can anyone just suggest me about Gaussian waveform generation?

• 'Wave' usually is a repetitive phenomenon, but a Gaussian shape is not. Usually, it describes a distribution of some random variable stretching from minus infinity to plus infinity (i.e. in unbounded space, no circular boundary condition). Aug 4, 2020 at 5:27
• You can use a squaring circuit (translinear, aka Barry Gilbert) first and then supply this to another BJT and investigate the collector current that results, using that to drive the scope's Y while the X is driven with a ramp circuit that feeds the squaring circuit, mentioned earlier. A unipolar combination can achieve half the Gaussian curve. A bipolar combination could achieve both halves. At least, it should be close enough. I haven't built this but given the basic Gaussian Function, this kind of circuit just falls to mind. Someone must have done this already.
– jonk
Aug 4, 2020 at 5:46
• Basically I want to say gaussian shaped signal @White3rd . The eqn you can think of for this as exp(-sin^2(t)). Where sine function is for periodic gaussian signal and exp(-t^2) is for gaussian shaped signal Aug 4, 2020 at 8:01
• @jonk I'll try that and will tell you the result.. Aug 4, 2020 at 8:03
• @ShafayetRahat I'm just looking at the exp() function itself and the squared x power. The rest is just constants, so to speak. Since the collector current is an exponential of the base-emitter voltage and since a translinear BJT/Gilbert arrangement squares the voltage, it seems a fit to me. I could sit down with Spice and work out the exact details. But the gist of it is there. The rest is details, I think.
– jonk
Aug 4, 2020 at 8:35

There is an EDABOARD thread ( https://www.edaboard.com/threads/hardware-design-for-gaussian-filter.264943/ ) that discusses exactly what you want, a Gaussian filter implemented with opamps. The post https://www.edaboard.com/threads/hardware-design-for-gaussian-filter.264943/#post-1139855 goes into details showing "a normalized 5th order Sallen Key filter". I attach a picture from this post to my answer: The post warns on issue of practical usefulness of this design:

If gaussian filter is not perfect the data may not be corrected I think. Dont know how gaussian is used in real gsm

I.e., discussed is the Gaussian modulation as used in GSM (see https://www.etsi.org/deliver/etsi_en/300900_300999/300959/08.01.02_60/en_300959v080102p.pdf). This kind of modulation is called Gaussian Minimum Shift Keying (GMSK), and it modulates the phase of the wave.

Bluetooth uses Gaussian Frequency Shift Keying. GFSK uses a Gaussian filter for pulse shaping. The practical designs use FPGAs or CORDIC hardware to calculate digital Gaussian data. I do not know if the analog Gaussian filters are ever used in BLE designs.

• Got it in my matlab simulation. Only have to change the load resister . Aug 13, 2020 at 17:08

classic Gaussian describes a random_process, such as myriad electrons bouncing between orbital energy levels.

The mass of un_synchronised energy changes results in the Gaussian randomness.

Opamps cannot by themselves GENERATE a random signal, but opamps can AMPLIFY a random voltage signal.

A 100,000 ohm resistor will give you 40 nanoVolts of random voltage for every cycle_per_second (hertz) of bandwidth. The voltage(s) of all the tiny slices of bandwidth will sum, as the Root Sum Square RSS, to make higher voltages and with frequency content from all the tiny slices you include.

[this explanation may seem mysterious, but we are in a new philosophical topic, crucial to system design yet using little or none of the words we are taught in circuit design; the nomenclature of randomness evolved in the 1800s, driven by attempts to understand behaviors of gases; Einstein benefitted from the new concepts of that era, and proposed the Photoelectric Effect in a paper, for which he got that Nobel Prize; in the 1920s, Johnson and Nyquist collaborated, to bring the behaviors of gases into the behaviors of circuits. Thus the idea of random_processes in circuits is less than 100 years old, but are crucial in establishing the lower end of dynamic range for our systems.]

From DC to 10,000Hz, our 100,000 ohm resistor's 40 nanoVolts per ccycle will increase b sqrt(10,000Hz), thus 40nanoVolts * sqrt(10,000) == 4 microVolts RMS.

The 1ppm voltage excursion (peaks) will be 6.2 RMS (aka sigma, or standard deviation), or6.2 * 4 = 24.8 microVolts peak (at the 0.0001 % occurrence).

So you can design a chain of amplifiers, working from DC to 10,000 or 20,000 Hertz, knowing the maximum (at th e0.0001% occuration) is 24.8 microvolts.

Using opamps with 10MHz UGGBW, you can achieve a accurate gain of 100X in one stage, producing 2.48 milliVOlts.

A second such gain stage gives you 248 milliVOlts, easy to view on an oscilloscope.

Again, the RMS will be 40 milliVolts, and the 6.2 sigma (standard deviation) will be 248 milliVolts.

Have fun.

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by the way, I used 10,000Hertz bandwidth bcause 100,000 ohm resistor and 10pF capacitance is 1uS time constant or 159,000Hertz bandwidth. Some opamps may have lots of Miller Capacitance, and easily reach 10pF Cinput at the Vin+ pin.

Note this gives you about 16X bandwidth, thus 4X more voltage. In that case, the Vout_1ppm will be about 1 volt, and the VOut_RMS will be 6x smaller at 160 milliVolts.

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https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise

Many simulation tools let you experiment with cascaded amplifiers and the noise floors of each amplifier.

I've recently used Signal Chain Explorer; build a chain of opamp amplifiers, select the ratio of resistors to set the gain of each, adjust the Unity Gain Bandwidth if the default is not what you want, then click "UPDATE".

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As Brian Drummond explains, what I've written is about random noise which with large # of un_synchronized spike_sources (electrons jumping around), will produce a RANDOM DISTRIBUTION, called a Gaussian Distribution.

There is also a Gaussian_Shaped pulse, likely best (accurately) synthesized from a lookup table and a DAC digital_analog_converter. Gaussian pulses have useful properties of lowest interference between Radio Channels.

• Thanks for your answer @analogsystemsrf. Can you plz share some kind of resource or link so that I can study them. Aug 4, 2020 at 7:55
• That's Gaussian distribution. There is also a "Gaussian" shaped pulse, and a Gaussian filter response. If I remember correctly the Laplace Transform of a Gaussian pulse is a Gaussian pulse.
– user16324
Aug 4, 2020 at 16:31
• Thanks for your answer. I am working on a project to generate a modulation of exp(-t2)*sinx which is a solution to another equation. And my professor don't want to use DAC. Aug 6, 2020 at 8:59
• So, if I understand you right you are suggesting to connect a chain of op amps with adjustment of resistor. Aug 6, 2020 at 9:07

I think the question is about wave-shape generation by an analog circuit. The DAC solution is quantized in both time and value.

1. Proposed Circuit to generate the Gaussian function, A. I. BAUTISTA-CASTILLO, ET AL., A CMOS MORLET WAVELET GENERATOR (2017)

The circuit uses an analog multiplier to obtain a squared value. The controlling input may be too complicated (triangular tuple).

2. But there are other circuits:

See also: Analog Gaussian Function Circuit: Architectures, Operating Principles and Applications, Alimisis et al. (2021) MDPI electronics.

3. There are other methods. If the input is a spiking pulse in time, the impulse response of a series of RC filters will start approaching a Gaussian: