Example Data Table
This example uses x[n] = 1, 2, 0, -1 with N = 4 and no normalization.
| n or k |
Input x[n] |
Expected Ck |
|Ck| |
| 0 |
1 |
2 + 0i |
2 |
| 1 |
2 |
1 - 3i |
3.162278 |
| 2 |
0 |
0 + 0i |
0 |
| 3 |
-1 |
1 + 3i |
3.162278 |
Formula Used
The forward discrete Fourier transform coefficient is:
Ck = alpha * sum from n = 0 to N - 1 of y[n] * e^(-j * 2 * pi * k * n / N)
Here, y[n] = (x[n] - mean) * w[n] when mean removal or a window is selected.
The scale alpha is 1, 1 / N, or 1 / sqrt(N). Magnitude is sqrt(real^2 + imag^2). Phase is atan2(imag, real). Frequency is k * sample rate / N.
How To Use This Calculator
- Enter real or complex samples in the sequence box.
- Leave N blank, or enter a custom transform length.
- Add the sample rate when frequency values are needed.
- Select direction, normalization, window, and decimal places.
- Press Calculate Ck to view results above the form.
- Use CSV or PDF export for saved reports.
Understanding DFT Ck Values
The discrete Fourier transform turns a finite sample list into frequency bins. Each bin is called Ck in many maths texts. It shows how much of one rotating wave exists in the input data. The calculator evaluates every bin with rectangular and polar results. It also shows phase, power, and bin frequency when a sample rate is given.
Why Ck Matters
Ck values help you compare hidden cycles in a sequence. A large magnitude means a strong repeating pattern at that bin. A small magnitude means that bin contributes little. The real part describes cosine strength. The imaginary part describes sine strength. The phase tells where the wave begins relative to the first sample.
Advanced Inputs
The calculator accepts real values and complex values. You may enter values like 4, -2, 3+5i, or 7-2j. A custom transform length can pad zeros or truncate samples. This is useful when you need a fixed number of bins. The window option can reduce edge jumps before the transform. Mean removal helps when a large constant level hides smaller cycles.
Reading The Output
The table lists k from zero to N minus one. The centered index shows negative frequency positions after the middle bin. Frequency equals k times the sample rate, divided by N. Magnitude shows the size of the coefficient. Phase is listed in radians and degrees. Power is the squared magnitude. The dominant bin marks the strongest coefficient after the selected scaling.
Good Practice
Start with a short known sequence. Check C0 first because it represents the sum or average level. Then compare mirrored bins when the input is purely real. They usually appear as complex conjugates. Increase N only when you need more bin positions. Zero padding does not add new information. It only samples the spectrum more closely.
Use In Maths Work
This tool is helpful for homework, signal examples, and manual checks. The export buttons create tables for notes and reports. Use the formula section to verify each term. Keep enough decimals for phase work. Round only at the final step when accuracy matters.
Remember that DFT bins depend on sample order. Changing one input value can change every Ck value. Always keep units and indexing consistent during review.
FAQs
What is Ck in a discrete Fourier transform?
Ck is the coefficient for frequency bin k. It contains a real part and an imaginary part. Together, they describe the size and phase of a rotating wave inside the sample sequence.
Can I enter complex samples?
Yes. Use i or j notation. Examples include 2+3i, -4j, 5-2i, and 7. Separate each sample with a comma, semicolon, or new line.
What happens when N is larger than my sample count?
The calculator adds zeros after your sequence. This is zero padding. It gives more bin positions, but it does not create new signal information.
What happens when N is smaller than my sample count?
The calculator uses the first N samples only. Later samples are ignored. Choose this only when you intentionally want a shorter transform.
Which normalization should I choose?
Use none for the common forward DFT form. Use 1 / N when you want average style coefficients. Use 1 / sqrt(N) for unitary energy scaling.
Why is C0 important?
C0 represents the constant part of the sequence. With no normalization, it equals the sum of prepared samples. With 1 / N normalization, it equals their average.
What does phase mean?
Phase shows the angle of a coefficient. It tells how the matching wave is shifted. The calculator displays phase in radians and degrees for easier checking.
Why use a window option?
A window can reduce sharp edge jumps in a finite sample block. This may reduce leakage across nearby bins. Use it when samples do not complete a clean cycle.