Fourier Analysis (friend) - 5ESC0

Notes by a friend for Signals II (DSP Fundamentals) - 5ESC0

• Topics: Fourier Transform · Sampling

Fourier Transform properties & proofs

Fourier Series to Fourier Transform

Fourier series - Wikipedia
Generalize Fourier series to Fourier transform (discrete to continuous spectrum):
(Continuous) Fourier Transform:
Relationship between Fourier series and Fourier transform (of one period):
Fourier transform of periodic signal:

Discrete Fourier transform:

Discrete Fourier transform of periodic signal:

Properties

• Additive:
• Homogeneous:
• Linear: additive + homogeneous
• Time invariant:
• Bounded input-Bounded output (BIBO) = Stability (absolutely integrable):

Link to original

Sampling

Topic: Sampling

Upsamling & Downsampling

Convert continuous signal to discrete sequence:
: Fourier transform of continuous signal
: Fourier transform of discrete samples
: Fourier transform of discrete sequence
: Fourier transform of decimated sample
: Fourier transform of decimated sequence

Sampling:

Sequencing:
Resampling:


Sampled at T1 → Downsample & Decimate by 2 → Reconstruct at T2=2T1
Sampled at T1 → Upsample & Interpolate by 2 → Downsample & Decimate 3 → Reconstruct at T2=3/2T1

Note

Zero Padding

Increase spectral resolution without new information (simply do more calculations with given samples)

Windowing

Window function - Wikipedia
Taking smaller samples of long signals → spectral leakage due to windowing
![[signals_window_rect.png|Rectangular window of ]]

Common window functions (most used is Hann window)

Relations between signal transformations:

Condition of Convergence

Dirichlet Invented this Function to Prove a Point - YouTube

Dirichlet

Weak:
Strong: in 1 period

• Finite number of extrema
• Finite number of discontinuities, each finite