Calculator Form
Example Data Table
Use this table to test common conversion cases.
| Case | Fs | Ts | Frequency | Pole s | Method | Expected Use |
|---|---|---|---|---|---|---|
| Audio tone sample | 8000 Hz | 0.000125 s | 440 Hz | -20 + j0 | Zero Order Hold | Signal sampling |
| Control filter | 100 Hz | 0.01 s | 5 Hz | -5 + j20 | Tustin | Controller design |
| Slow sensor | 10 Hz | 0.1 s | 1 Hz | -1 + j0 | Backward Euler | Stable simulation |
Formula Used
Sampling Period
Ts = 1 / Fs
Sample Index
n = round(t / Ts) and t_sample = nTs
Sinusoidal Signal Sampling
x[n] = offset + A sin(2π f nTs + phase)
Continuous Pole to Discrete Pole
z = e^(sTs). If s = a + jb, then z = e^(aTs)(cos(bTs) + j sin(bTs)).
Zero Order Hold for First Order Model
H(s) = K / (tau s + 1)
y[n] = e^(-Ts/tau)y[n-1] + K(1 - e^(-Ts/tau))u[n-1]
Forward Euler
y[n] = (1 - Ts/tau)y[n-1] + K(Ts/tau)u[n-1]
Backward Euler
y[n] = (tau / (tau + Ts))y[n-1] + (KTs / (tau + Ts))u[n]
Bilinear Tustin
s = (2 / Ts)(1 - z^-1) / (1 + z^-1)
How to Use This Calculator
- Enter the sampling rate or enter the sample period.
- Choose the continuous signal type.
- Add amplitude, frequency, phase, offset, and time values.
- Enter the continuous pole real and imaginary parts.
- Add gain and time constant for the first order model.
- Select the preferred discretization method.
- Press the calculate button.
- Review the result shown above the form.
- Download the CSV or PDF report when needed.
Continuous Time to Discrete Time Conversion Guide
What the Calculator Does
A continuous time signal exists at every instant. A discrete time signal exists only at selected sample points. This calculator connects both worlds. It converts time, signal value, frequency, pole location, and first order model coefficients into a sampled form. Engineers use this step when they move from analog equations to digital controllers, filters, simulations, and data systems.
Why Sampling Period Matters
The sampling period is the time gap between two samples. A smaller sampling period gives more detail. A larger sampling period gives fewer points and may hide fast changes. The sampling rate is the inverse of the sampling period. When the sampling rate is high, the discrete model can follow the continuous model more closely. When it is low, aliasing and poor dynamic response can appear.
Signal Sampling
The calculator evaluates the continuous signal at the nearest sample index. For a sine signal, it uses amplitude, frequency, phase, and offset. For an exponential signal, it uses the alpha value. Step and ramp options are also included. This makes the tool useful for signal processing, measurement design, education, and quick engineering checks.
Pole Mapping
Poles describe system behavior. A continuous pole is usually written as s. A discrete pole is usually written as z. The core mapping is z equals e raised to s times Ts. Stable continuous poles with negative real parts usually map inside the unit circle. That is important because discrete systems are stable when their poles remain inside that circle.
Discretization Methods
Zero Order Hold assumes the input stays constant during each sample interval. It is widely used for digital control. Forward Euler is simple and fast, but it can become unstable when the sample period is too large. Backward Euler is more stable for many simulations. Tustin, also called bilinear conversion, maps frequency behavior in a balanced way and is popular for filter work.
First Order Model Conversion
The first order model uses gain and time constant. The gain controls final response size. The time constant controls speed. The calculator creates difference equations for several methods. These equations can be placed inside a digital controller, spreadsheet, embedded device, or simulation loop. The output also shows the model pole created by the time constant.
Aliasing and Nyquist Limit
Aliasing happens when a signal frequency is too high for the chosen sampling rate. The Nyquist frequency is half of the sampling rate. A clean sampled design normally keeps important signal content below that limit. The calculator reports an estimated alias frequency so the user can quickly see whether a selected frequency may cause confusion after sampling.
Practical Tips
Use a sampling rate that is much higher than the fastest important signal behavior. Many control systems use rates ten to twenty times faster than the dominant bandwidth. For audio and communication work, follow stricter design rules. Always compare the method results. A method that works for one model may not be ideal for another model.
Exporting Results
The CSV export is useful for spreadsheets and records. The PDF export is useful for reports and sharing. Both options help preserve the selected inputs, calculated values, and method comparison. This makes the calculator helpful for homework, documentation, audits, design reviews, and repeated engineering tasks.
FAQs
1. What is continuous time?
Continuous time means the signal or model is defined for every instant. Analog circuits, physical systems, and differential equations often use continuous time.
2. What is discrete time?
Discrete time means the signal is known only at sample points. Digital controllers, sampled data systems, and computer simulations usually use discrete time.
3. What is sampling period?
Sampling period is the time between two samples. It is written as Ts. A smaller value means more samples are taken each second.
4. What is sampling rate?
Sampling rate is the number of samples taken each second. It is written as Fs. It equals one divided by the sampling period.
5. What does z = e^(sTs) mean?
It maps a continuous pole from the s-plane into a discrete pole in the z-plane. This helps compare stability and dynamic behavior.
6. Which method should I choose?
Zero Order Hold is common for control systems. Tustin is common for filters. Backward Euler is useful for stable simulations. Forward Euler is simple but less robust.
7. What is Zero Order Hold?
Zero Order Hold assumes the input remains constant during each sample interval. It matches many digital control systems that hold actuator commands.
8. What is Forward Euler?
Forward Euler approximates derivatives using a forward step. It is easy to apply but may be inaccurate or unstable with large sample periods.
9. What is Backward Euler?
Backward Euler uses the current input in the update. It often gives better stability than Forward Euler for simple dynamic models.
10. What is Tustin conversion?
Tustin conversion uses a bilinear substitution for s. It is popular because it preserves useful frequency behavior in many digital filter designs.
11. What is aliasing?
Aliasing occurs when high frequency content appears as a lower frequency after sampling. It happens when sampling is too slow for the signal.
12. What is Nyquist frequency?
Nyquist frequency is half the sampling rate. Important signal frequencies should usually stay below this limit to reduce aliasing risk.
13. How is stability checked?
A continuous pole is stable when its real part is negative. A discrete pole is stable when its magnitude is less than one.
14. Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable report with calculated values and method results.