Transforming Signals in Time-Domain into Signals in Frequency-Domain Fourier Series (FS) Fourier Transform (FT) Fast Fourier Transform (FFT) Discrete Time Fourier Transform (DTFT) Z-Transform My own reserach paper that combines FS and Z-transform. Hülya Yalçın ©
Fourier Series (FS) The orthogonal bases functions for Fourier Series are cos(nt) and sin(nt). Hülya Yalçın ©
Discrete Fourier Transform etc Well, there are also Fourier Transform Discrete Fourier Transform etc How all these things are related? Let’s first go through Fourier Transform. Hülya Yalçın ©
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Fourier Transform (FT) Fourier Series (FS) The orthogonal bases functions for Fourier Series are cos(nt) and sin(nt). Fourier Transform (FT) The orthogonal bases functions for Fourier Transform are e^(iwt). Hülya Yalçın ©
Going from Fourier Series to Fourier Transform Hülya Yalçın ©
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Understanding how Fourier Series (FS) goes to Fourier Transform (FT)... In time domain, making a signal periodic corresponds to convolving it with a COMB function. The fourier transform of a COMB function is again a COMB function. So, in time domain it corresponds to sampling the frequency domain signal! Hülya Yalçın ©
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Famous Fourier Transforms Sine wave Delta function
Famous Fourier Transforms Gaussian Gaussian
Famous Fourier Transforms Sinc function Square wave
Famous Fourier Transforms Sinc function Square wave
Famous Fourier Transforms Exponential Lorentzian
Trigonometric Fourier series: example Hülya Yalçın ©
Complex Exponential Fourier Series Hülya Yalçın ©
Complex Exponential Fourier Series Hülya Yalçın ©
Symmetry Properties for real signals Hülya Yalçın ©
Amplitude and phase spectra Hülya Yalçın ©
Fourier Transform Hülya Yalçın ©
Recap: Fourier Series Hülya Yalçın ©
Spectral content of aperiodic signals: the Fourier transform Hülya Yalçın ©
Example: rectangular pulse Hülya Yalçın ©
Inverse Fourier Transform Hülya Yalçın ©
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Properties of Fourier Transform Hülya Yalçın ©
Properties of Fourier Transform: Linearity Hülya Yalçın ©
Properties of Fourier Transform: Time-shift Hülya Yalçın ©
Multiplication by a complex exponential Multiplication by a cosine Hülya Yalçın ©
Convolution in time-domain Hülya Yalçın ©
Parseval’s Theorem Hülya Yalçın ©
Duality Example Hülya Yalçın ©
Generalized Fourier Transform Hülya Yalçın ©
Fourier Transform of cosine Hülya Yalçın ©
Fourier transform of a periodic signal Hülya Yalçın ©
Bandlimited and timelimited signals Hülya Yalçın ©
Frequency response of LTI systems Hülya Yalçın ©