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- Basis Function
The mathematical definition of the basis functions used in harmonic analysis is essential for decomposing signals into their frequency components via the Fourier transform, and, in practice, signals are expanded in periodic sine–cosine bases to enable detection of a signal’s presence and estimation of its parameters.
k : Index of the frequency component (harmonic). In sampled signals, components are considered up to the Nyquist frequency, i.e., k = N/2 (for even N).
N : Number of samples. It denotes the uniform sample count used for the DFT.
T : Sampling interval (sample period), i.e., the time between adjacent samples. (Sampling rate fs = 1/T.)
NT : Total duration of the observation interval.
n : Discrete-time sample index; takes N points from 0 to N−1.
(2π/N)kn : The phase (in radians) of the k-th frequency component at sample index n (i.e., with angular frequency ω_k = 2πk/N, the phase is ωkn).
- DFT Sampling Essentials
Orthogonality of sinusoids: Uniform sampling over an integer number of periods yields orthogonal discrete sine/cosine sequences.
Top (True-even): The samples cover one full period; the last sample n = 15 is the end of that period (for N = 16).
Bottom (DFT-even): The last sample is omitted; at n = 16, the periodic extension repeats the n = 0 value, completing the period.
- Two common FT
1. Hz (Frequency f) Based
2. rad/s (Angular Frequency ω) Based
- The Definition of the DFT
X[k] represents how much x[n] overlaps (inner product)
with the k-th basis function.
The dot product can be used to test of two vectors are perpendicular.
If they are, the component of one along the other must be zero so the dot product must be zero.
- The DFT of a length-N signal x[n] :
k : Index of the frequency component (harmonic). In sampled signals, components are considered up to the Nyquist frequency, i.e., k = N/2 (for even N).
N : Number of samples. It denotes the uniform sample count used for the DFT.
n : Discrete-time sample index; takes N points from 0 to N−1.
Note : N = 32, fs = 32 KHz, k = 1,2,3
- Meaning of k0 (DFT bin index)
The DFT bin index k0 corresponds to the normalized frequency in the time domain. For N samples:
To convert it to actual frequency in hertz (Hz), multiply by the sampling frequency fs :
If the sampling frequency is fs = 32 kHz:
• k0 = 1 → f = 1/16 × 16,000 =1,000 Hz
• k0 = 2 → 2,000 Hz
• k0 = 3 → 3,000 Hz
If the sampling frequency is fs = 16 kHz:
• k0 = 1 → 500 Hz
• k0 = 2 → 1,000 Hz
• k0 = 3 → 1,500 Hz
- Why Leakage Shows Up (or Not): A One-Line Rule with the DFT
N : Total record length / DFT size (window length). Example: N = 1024 samples.
N0 : Period in samples (samples per cycle).
Time (Coherent)
: N = 32, N0 = 16. One period N0 is shaded. Because N is an integer multiple of N0, tiling the record end-to-end introduces no discontinuity.
Time (Non-Coherent)
: N = 28, N0 = 16. The record length is not an integer multiple of the period, so tiling creates an edge discontinuity.
Frequency (Coherent)
: Using the DFT definition
the energy sits only at the two bins ±k0 → no spectral leakage.
Frequency (Non-Coherent)
: With the same DFT definition, energy spreads over neighboring bins → spectral leakage.
The one-line rule
• No leakage (Integer) : N = L N0 ⟺ f0/Δf ∈ Z.
• Leakage (Not Integer) : N ≠ L N0 ⟺ f0/Δf ∉ Z.
( Notes : k0 = f0/Δf, Δf = fs/N )
Integer number of periods in the record → edges match → no leakage.
Non-integer number of periods → edge discontinuity → spectral leakage.
Root cause: a mismatch between the DFT’s implicit periodic extension and the signal’s actual period.
Window functions: taper the signal toward zero at the record edges, softening the discontinuity and reducing sidelobe leakage (at the cost of a wider main lobe).
- CTFT vs. DTFT — Sampling, Periodic Spectra, and Aliasing
CTFT : The Continuous-Time Fourier Transform
DTFT : The Discrete-Time Fourier Transform
CTFT
: The CTFT of a continuous-time signal f(t). Time is continuous and the spectrum is generally aperiodic.
DTFT
: The DTFT of the samples x[n] = f(nT) (equivalently, the CTFT of the impulse-sampled signal fs(t) = ∑nf(nT)δ(t - nT). It’s a sum, and the spectrum is periodic with period ωs = 2π/T.
T : The sampling period (in seconds). This means we observe the continuous-time signal f(t) only at equally spaced instants t = 0, ±T, ±2T,…. The corresponding sampling angular frequency is ωs = 2π/T (rad/s).
n : The integer sample index (…,-2,-1,0,1,2,…). The actual sampling time is t = nT, and the sampled value is x[n] = f(nT).
Relationship (Poisson Summation) :
: In words: DTFT is CTFT’s spectrum F(ω) repeated every ωs and scaled by 1/T.
If F(ω) is band-limited to ∣ω∣ < ωs / 2, you can perfectly recover F(ω); otherwise the replicas overlap and aliasing occurs.
The original spectrum F(ω) is replicated (periodically repeated) at intervals of ωs. Therefore Fs(ω) is a periodic function with period ωs = 2π/T.
- Continuous-time Signal f(t)
- Impulse train s(t), T = 0.1s
- Sampled Signal fs(t), T = 0.1s
- Impulse train s(t), T = 0.25s
- Sampled Signal fs(t), T = 0.25s
- Discrete-time sequence x[n] = f(nT), T = 0.1s
- Discrete-time sequence x[n] = f(nT), T = 0.25s
Why is
nonzero only at t = nT and zero elsewhere?
Single-impulse property.
The delta is not an ordinary function with pointwise values; it is a distribution whose entire “mass” (area 1) is concentrated at the single point t = a. For any smooth g(t),
which is called the sifting property.
Time shift.
δ(t − nT) is just δ(t) shifted by nT to the right. Therefore the spike occurs only at t = nT, and it behaves like zero elsewhere. Summing over all integers n places spikes at every multiple of T.
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