01_FFT_Basic

The Fourier Transform is a mathematical operation that transforms a time-domain signal into its frequency-domain representation. It decomposes a signal into its constituent sinusoidal components, allowing us to analyze the frequency content of the signal. For a continuous-time signal x(t), the Fourier Transform is defined as:

Here,  X(ω) represents the frequency spectrum of the signal, and ω(=2πf) is the angular frequency. This transformation is fundamental in signal processing, communications, and many areas of engineering and physics.

To understand the Fourier transform, we will analyze a signal in the time domain. As an example, consider the signal x(t) = cos(ωot), which can be expressed using Euler's formula as:

 - From Rectangular Window to Dirac Delta function δ(α) 

1. First integral (with respect to t)
  : The result of the integral of ejαt over the interval  [−T,T] depends on the value of α.
    Integrating t gives the function IT(α), a function of α.

2. Second integral (with respect to α)
  : Now we compute the area (the α-Integral) of IT(α) :

Here π appears because of the Dirichlet integral

 - Sinc Function 

 -  KT(𝛼) = 2sin(𝛼T)/𝛼 

The key point is that δ(α) (and 2π δ(α)) is not an ordinary function but a distribution. You can’t plot it as a single curve; instead, it is defined by its action on smooth test functions φ(α):

 In other words, δ has no value “by itself”—it only has meaning under an integral against a smooth φ.

The windowed Fourier kernel IT(α) = 2T sinc(αT) becomes taller and narrower as T → ∞ while keeping its total area equal to 2π; in the distributional sense IT(α) → 2π δ(α), with main-lobe width approximately 2π/T. This is why IT​ serves as an “approximate identity” that concentrates mass at α = 0.

Why do we perform this second integral?

In the distributional interpretation, to confirm the normalization

It shows that IT(α) = 2T sinc(αT) becomes a taller-and-narrower kernel as T → ∞ while keeping its total area equal to 2π. Computing that total area is exactly what the second α-integral does. So we are not repeating the same integral; we first integrate over t to obtain the function, then over α to verify its area (normalization). 

(Here sinc(x) = sinx/x)

 Note : 

• Sinc function : A function that maps a real number x to “sin x divided by x.” In signal processing, a “normalized” version maps x to “sin of pi times x divided by pi times x.” It’s just a function.

• Dirichlet integral : A fact about the sinc curve: the total signed area under it across all real x equals pi. This is a single numerical value, not a function.

 - The Continuous-Time Fourier Transform (Cosine Wave) 

Now we compute the area (the α-Integral) of IT(α) :

• Each delta’s area (weight) is π, so the two together sum to 2π.

    : For a cosine windowed on the rectangular interval [-T, T], the first zero of one lobe (e.g., the +ω0​ lobe) occurs at

• Radians : ω = ω0 ± π/T

• Hertz : f = f0 ± 1/2T

Note: Because a cosine is the sum of two lobes, |XT| may not be exactly zero at those points, but they are the theoretical first zeros of that lobe.

 - Cosine: CTFT vs DFT (Rectangular Window) 

 - Cosine: CTFT vs DFT (Non-Coherent) 

 - Cosine: CTFT vs DFT (Coherent) 

• The exact condition (FFT view)

With sample rate fs​ and record/FFT length N, a tone at frequency f0​ lands exactly on an FFT bin when
(• Zero leakage (with rectangular window): requires f0 = k fs /N.

• Non-integer cycles (f0 ≠ k fs /N) → energy spreads across many bins via the Dirichlet kernel.)

Equivalently, the record time is Trec = N/fs ​, and the number of cycles in the record

is an integer.

Another way to say it: the samples-per-cycle is S = fs /f0 ​, and if N is a multiple of S (i.e., N = mS), there’s no leakage.

With a rectangular window, a real cosine/sine places energy only in the two symmetric bins k and N−k;

all other bins are zero (up to numerical error). The phase ϕ only changes the complex coefficients (sign/angle); it does not affect leakage.

CTFT on a Finite Interval: Why the Spectrum Looks Sinc-Shaped

In a finite observation window, the continuous-time Fourier transform (CTFT) of a sinusoid no longer appears as ideal impulses. Time truncation multiplies the signal by a rectangular window wT(t) for ∣t∣ ≤ T (and 0 otherwise). By Fourier duality, multiplication in time becomes convolution in frequency, so the two impulses at ±ω0 are convolved with the window’s spectrum WT(ω) = 2sin(ωT) /ω. The result is

This produces sinc-shaped main lobes centered at ±ω0 with side lobes that decay roughly as 1/ω. The main-lobe width to the first zero is about 2π/T: as T increases, the lobes narrow and the peaks grow, tending to ideal impulses only in the limit T → ∞. Thus, even when the window contains an integer number of cycles, the CTFT magnitude versus ω\omegaω looks spread (sinc-like) because of the windowing, not because of any bin mismatch.

• Signal Frequency : 5 Hz,  

• FFT Duration : 1 s,  

• Sampling Frequency : 20 Hz and 11 Hz (>Nyquist) / 6 Hz (<Nyquist)

• Signal Frequency : 5 Hz,  

• FFT Duration : 10 s (10 times the duration of 1 second),  

• Sampling Frequency : 20 Hz and 11 Hz (>Nyquist) / 6 Hz (<Nyquist)