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Fourier, bandwidth, filter

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• 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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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

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a. Low freq. signal; b. High freq. signal; c. Medium freq. signal

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Bandlimited signal

7 / 13Symmetry Properties to Fourier Transform Discrete-Time

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• Aperiodic discrete-time signal x(n) with Fourier transform:

• DFT ; k = 0, 1, 2, …, N-1

• IDFT ; n = 0, 1, 2, …, N-1

-n

nj-e )n(x)(X

1-N

0n

kn/Nj2-e )n(x)k(X

1-N

0k

kn/Nj2e X(k) N

1)n(x

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Penjumlahan sinyal

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The Gibb’s Effect

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The Gibb’s Effect

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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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