Summary of Bode Analyzer using STM32F407 Discovery board
This article explains frequency domain analysis using Bode plots, detailing a method to generate sine waves via a 2048-point lookup table and linear interpolation for variable frequencies. It highlights the importance of a fixed 200KHz sampling rate for digital signal processing and compares custom generation against external DDS chips like the AD9833, emphasizing that knowing both sine and cosine values simplifies magnitude and phase detection without needing separate 90-degree phase shifters.
Parts used in the Frequency Domain Analysis Project:
- Floating point device
- Digital-to-Analog Converter (DAC)
- Analog-to-Digital Converter (ADC)
- Sine table (2048 points)
- Linear interpolation technique
- External DDS chip (AD9833) - mentioned as an alternative
Introduction
Frequency domain analysis is very important to know more about any system/ circuit or transfer function that we deal with. And the first thought comes to our mind about frequency domain analysis is “Bode plot“. Bode plot is a combination plot of magnitude and phase difference of output vs. input of a cosine wave of single frequency, when it is applied to a circuit.
Each of these cosine wave (signal) are applied once at a time and the frequency (in Hz or rad/sec) is increased in linear order (called chirp). All the while the amplitude of sine wave is kept constant (at say 1V_peak). While the frequency can be increased in small steps (1Hz/ sec), the time required to complete the entire frequency range can be quite big. Hence, it is preferred to choose 20 frequencies/ decade. For example, I had chosen 05Hz, 1Hz, 1.5Hz, 2Hz, 2.5Hz… 10Hz.
Generation (output) and sampling (input or feedback) of Sine wave
Generation of cosine wave (and sine) waveform is pretty straight forward, using floating point device and DAC. Be aware that using the sin or cos function from standard “math.h” library takes a lot of time (because, it is designed to meet a certain resolution, by iterative process). Instead, I am using a sine table (of 2048 points of amplitude, over 0 to 2*pi rad). I kept the “sampling” frequency (for both DAC and ADC) at 200KHz. Keeping a fixed frequency is vital to any digital signal processing algorithm.
Now the complication, if different output frequencies are needed: –
- Keeping sampling frequency of 200KHz and output of 1Hz = 200,000 points in 1 sine table.
- Keeping sampling frequency of 200KHz and output of 10KHz = 20 points in 1 sine table.
Obviously, a straight forward approach of a single sine table will not suffice. Hence, I use a linear interpolation technique. If the angle is such that direct output is available from the 2048 sine table, the the value is directly used. For values of angle in between, the amplitude is arrived using linear interpolation of 2 nearby values. For higher frequencies, the points in sine table are skipped (and interpolated). A function is written, which take care of all these logic and outputs the amplitude for any angle needed.
I could have used external DDS chips such as AD9833, but knowing both the sine and cosine of signal being generated, makes the magnitude and phase detection simpler (explained later). If I had used externally (DDS) generated signal, then I would have known the “frequency” of output signal (by knowing the SPI command of frequency that I send to AD9833), but knowing the phase is impossible, without sampling. And then, we would have needed separate means to get 90-deg phase shifted waveform.
- What is a Bode plot?
A combination plot of magnitude and phase difference of output versus input for a cosine wave of single frequency applied to a circuit. - How should frequency be increased during analysis?
Frequency is increased in linear order, often called a chirp, with a preference for choosing 20 frequencies per decade. - Why is a standard math library sine function avoided?
The sin or cos functions from standard libraries take too much time because they use an iterative process designed for specific resolution. - What sampling frequency is used for the DAC and ADC?
A fixed sampling frequency of 200KHz is kept for both DAC and ADC to ensure vital conditions for digital signal processing algorithms. - How does the system handle different output frequencies with a fixed sampling rate?
The system uses a linear interpolation technique where points in the sine table are skipped and interpolated based on nearby values. - Why might one choose custom sine wave generation over external DDS chips?
Knowing both the sine and cosine of the generated signal makes magnitude and phase detection simpler compared to external signals where phase is hard to know. - What problem arises when using an external DDS chip like AD9833?
While frequency is known via SPI commands, determining the phase is impossible without sampling, requiring separate means to get a 90-degree phase shifted waveform.
