Transforming Signals in Time-Domain into Signals in Frequency-Domain

... konulu sunumlar: "Transforming Signals in Time-Domain into Signals in Frequency-Domain"— Sunum transkripti:

1 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 ©

2 Fourier Series (FS) The orthogonal bases functions for Fourier Series are cos(nt) and sin(nt). Hülya Yalçın ©

3 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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9 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 ©

10 Going from Fourier Series to Fourier Transform
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30 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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44 Famous Fourier Transforms
Sine wave Delta function

45 Famous Fourier Transforms
Gaussian Gaussian

46 Famous Fourier Transforms
Sinc function Square wave

47 Famous Fourier Transforms
Sinc function Square wave

48 Famous Fourier Transforms
Exponential Lorentzian

49 Trigonometric Fourier series: example
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50 Complex Exponential Fourier Series
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51 Complex Exponential Fourier Series
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52 Symmetry Properties for real signals
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53 Amplitude and phase spectra
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54 Fourier Transform Hülya Yalçın ©

55 Recap: Fourier Series Hülya Yalçın ©

56 Spectral content of aperiodic signals: the Fourier transform
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57 Example: rectangular pulse
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58 Inverse Fourier Transform
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60 Properties of Fourier Transform
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61 Properties of Fourier Transform: Linearity
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62 Properties of Fourier Transform: Time-shift
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63 Multiplication by a complex exponential
Multiplication by a cosine Hülya Yalçın ©

64 Convolution in time-domain
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65 Parseval’s Theorem Hülya Yalçın ©

66 Duality Example Hülya Yalçın ©

67 Generalized Fourier Transform
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68 Fourier Transform of cosine
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69 Fourier transform of a periodic signal
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70 Bandlimited and timelimited signals
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71 Frequency response of LTI systems
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