A great website for reference: https://www.dspguide.com/ch1.htm
- codeblock: https://filehippo.com/download_codeblocks/
- gnuplot: https://sourceforge.net/projects/gnuplot/files/gnuplot/
All the following summary are reference from the website: https://filehippo.com/download_codeblocks/
The sample-and-hold (S/H) is required to keep the voltage entering the ADC constant while the conversion is taking place. It is allowed to change only at periodic intervals, at which time it is made identical to the instantaneous value of the input signal. Changes in the input signal that occur between these sampling times are completely ignored. That is, sampling converts the independent variable (time in this example) from continuous to discrete.
Quantization converts the dependent variable (voltage in this example) from continuous to discrete. Any one sample in the digitized signal can have a maximum error of ±? LSB (Least Significant Bit)
The sampling theorem is an analysis of what happens in the frequency domain during digitization. When a signal input to a system, if we want to know what the signal we get, then we need to do sampling. It will be great if we can get infinite sample, however, that is imposible and waste of resources. According to Nyquist Theorem, the sampling frequency should be two times larger than the maximum frequency of the input signal in order to reconstruct the exact signal (ie. frequencies above one-half the sampling rate are lost)
f_s >= 2*f_max
where f_s is sampling frequency and f_max is the maximum frequency of the input signal.
Nearly all DACs operate by holding the last value until another sample is received. This is called a zeroth-order hold, the DAC equivalent of the sample-and-hold used during ADC. (A first-order hold is straight lines between the points, a second-order hold uses parabolas, etc.). The zeroth-order hold produces the staircase appearance. However, in the frequency domain, the zeroth-order hold results in high frequency amplitude reduction and will have the shape like sin(πx)/(πx) (called sinc function)
Three types of analog filters are commonly used: Chebyshev, Butterworth, and Bessel. The complexity of each filter can be adjusted by selecting the number of poles. The more poles in a filter (by cascading the building block showing below), the more electronics it requires, and the better it performs. The following is a common building block for analog filter design, the modified Sallen-Key circuit.
The performance comparision is in the following. The Chebyshev optimizes the roll-off, the Butterworth optimizes the passband flatness, and the Bessel optimizes the step response.
The particular op amp use isn't critical, as long as the unity gain frequency is more than 30 to 100 times higher than the filter's cutoff frequency. This is an easy requirement as long as the filter's cutoff frequency is below about 100 kHz.
However, integrated circuit is difficult to make resistors directly in silicon, so switched capacitor filter come to the rescue. Notice that, in most systems, the frequency band between about 0.4 and 0.5 of the sampling frequency is an unusable wasteland of filter roll-off and aliased signals.
There is a strong trend in electronics to replace analog circuitry with digital algorithms. Multirate techniques, using more than one sampling rate in the same system.
This usually implies lowpass-filtering a signal, then throwing away some of its samples. First, pass the voice signal through a simple RC low-pass filter and sample the data at 64 kHz. The resulting digital data contains the desired voice band between 100 and 3000 hertz, but also has an unusable band between 3 kHz and 32 kHz. Second, remove these unusable frequencies in software, by using a digital low-pass filter at 3 kHz. Third, resample the digital signal from 64 kHz to 8 kHz by simply discarding every seven out of eight samples, a procedure called decimation. The resulting digital data is equivalent to that produced by aggressive analog filtering and direct 8 kHz sampling.
It is the process of upsampling followed by filtering. Upsampling is the process of inserting zero-valued samples between original samples to increase the sampling rate. (This is called “zero-stuffing”.) Upsampling adds to the original signal undesired spectral images (distorted) which are centered on multiples of the original sampling rate. The filtering removes the undesired spectral images.A youtube tutorial video for your reference: https://www.youtube.com/watch?v=vp4nKygufEc .
The 8 kHz data is pulled from memory and converted to a 64 kHz sampling rate, a procedure called interpolation. This involves placing seven samples, with a value of zero, between each of the samples obtained from memory. The resulting signal is a digital impulse train, containing the desired voice band between 100 and 3000 hertz, plus spectral duplications between 3 kHz and 32 kHz. Everything above 3 kHz is then removed with a digital low-pass filter. After conversion to an analog signal through a DAC, a simple RC network is all that is required to produce the final voice signal.
The following is a block diagram of a typical delta modulator. The analog input is a voice signal with an amplitude of a few volts, while the output signal is a stream of digital ones and zeros. A comparator decides which has the greater voltage, the incoming analog signal, or the voltage stored on the capacitor. This decision, in the form of a digital one or zero, is applied to the input of the latch. At each clock pulse, typically at a few hundred kilohertz, the latch transfers whatever digital state appears on its input, to its output. This latch insures that the output is synchronized with the clock, thereby defining the sampling rate.
If the output is a digital one, the switch connects the capacitor to a positive charge injector which increases the voltage on the capacitor by a fixed amount. If the output is a digital zero, the switch is connected to a negative charge injector. This decreases the voltage on the capacitor by the same fixed amount.
