# 1 / 13 Fourier, bandwidth, filter. 2 / 13 The important roles of Fourier series and Fourier transforms: –To analysis and synthesis signals in frequency.

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1 / 13 Fourier, bandwidth, filter

2 / 13 The important roles of Fourier series and Fourier transforms: –To analysis and synthesis signals in frequency domain Signals can be categories as periodic continuous time signal, aperiodic continuous time signal, periodic discrete time signal, and aperiodic discrete time signal

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4 / 13 BANDWIDTH The Concept of Bandwidth –Low freq. signal: power spectrum density (psd) or energy density spectrum is concentrated at 0 Hz. –High freq. signal: psd or energy density spectrum is concentrated at high frequency. –Medium freq. signal or bandpass signal: psd or energy density spectrum is concentrated between low freq. to high freq. Bandwidth signal: range freq. where psd is concentrated. –Eg. If 95% of psd in F1  F  F2, then 95% of signal bandwidth is F2 – F1. Bandlimited signal: spectrum = 0 at |F|  B, B is bandwidth. The half-power bandwidth, which is perhaps the most common bandwidth definition: the interval between freq. at which psd dropped to half of its maximum value of power, or 3 dB below the peak value

5 / 13 a. Low freq. signal; b. High freq. signal; c. Medium freq. signal

6 / 13 Bandlimited signal

7 / 13 Symmetry Properties to Fourier Transform Discrete-Time

8 / 13 Aperiodic discrete-time signal x(n) with Fourier transform: DFT ; k = 0, 1, 2, …, N-1 IDFT ; n = 0, 1, 2, …, N-1

9 / 13 Penjumlahan sinyal

10 / 13 The Gibb’s Effect

11 / 13 The Gibb’s Effect

12 / 13 Filter Noise reduction the moving average filter very good for many applications, it is optimal for a common problem, reducing random white noise while keeping the sharpest step response

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