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Systems (filters)

Systems (filters). Non-periodic signal has continuous spectrum Sampling in one domain implies periodicity in another domain. Periodic sampled signal has always discrete and periodic spectrum. time frequency. One way of “signal processing”. PROCESSING.

henry-levine
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Systems (filters)

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  1. Systems(filters)

  2. Non-periodic signal has continuous spectrum Sampling in one domain implies periodicity in another domain Periodic sampled signal has always discrete and periodic spectrum time frequency

  3. One way of “signal processing” PROCESSING

  4. Frequency response Linear system k*input input k*output output system system frequency response = output/input

  5. deciBel [dB] Log-log frequency response

  6. Memoryless system (amplifier) 2x Output at time t depends only on the input at time t Frequency response of the system Magnitude (dB) phase 3 0 frequency frequency 1 10 100 1000 1 10 100 1000

  7. System with a memory (differentiator) out in Frequency response of the differentiator (high-pass filter) 0 t0 0 t0 time time - 1 sample delay

  8. System with a memory (integrator) out in Frequency response of the integrator (low-pass filter) 0 t0 0 t0 time time + 1 sample delay

  9. TD - const delay TD Comb filter TD=T1 Frequency response of the system TD=3T2 magnitude TD=5T3 1 e.t.c e.t.c. 0 frequency 3/TD 5/TD 1/TD

  10. linear system nonlinear system output output input input

  11. noisy system noise

  12. Pulse train 10 ms 2 ms Its magnitude spectrum

  13. 2 ms 10 ms 20 ms

  14. T • For a single pulse, • the period becomes infinite • the sum in Fourier series becomes integral THE LINE SPECTRUM BECOMES CONTINUOUS

  15. Dirac impulse contains all frequencies 1/dt time dt frequency 0 Dirac impulse Impulse response Frequency response Fourier transform system time time frequency Fourier transform of the impulse response of a system is its frequency response!

  16. Sinusoidal signal (pure tone) Its spectrum T 1/T time [s] frequency [Hz] Truncated sinusoidal signal Its spectrum DT ?

  17. Truncated signal time [s] Infinite signal multiplied by square window Multiplication in one (time) domain is convolution in the dual (frequency) domain

  18. tp Pulse train ∞ ∞ - 10 ms 2 ms Its magnitude spectrum 0 1/tp 2/tp 3/tp frequency line spectrum with |sinc| envelope continuous |sinc| function f = 1/2 103 =500 Hz

  19. Convolution of the impulse with any function yields this function Spectrum of an infinite 1 kHz sinusoidal signal Truncated 1000 frequency [Hz]

  20. Dt = ∞ Dt = 100 ms Dt = 13 ms 850 Hz 0

  21. Narrow-band (high frequency resolution) system Wide-band (low frequency resolution) system frequency time

  22. Narrow-band (high frequency resolution) Broad-band (low frequency resolution) Long impulse response (low temporal resolution) Short impulse response (high temporal resolution)

  23. Time-Frequency Compromise • Fine resolution in one domain (df-> 0 or dt->0) requires infinite observation interval and therefore pure resolution in the dual domain (DT-> or DF-> ) • You cannot simultaneously know the exact frequency and the exact temporal locality of the event • infinitely sharp (ideal) filter would require infinitely long delay before it delivers the output

  24. signal is typically changing in time (non-stationary) time short-term analysis: consider only a short segment of the signal at any given time DT DT to analysis the signal appear to be periods with the period DT

  25. Non-stationary turns into periodic

  26. Discrete Fourier Transform Discrete and periodic in both domains (time and frequency)

  27. Short-term Discrete Fourier Transform

  28. Signal multiplied by the window Spectrum of the signal convolves with the spectrum of the window

  29. frequency time time

  30. frequency time

  31. Analysis window 5 ms Analysis window 50 ms 5 frequency [kHz] 0 0 time [s] 1.2 0 time [s] 1.2 log amplitude frequency frequency

  32. frequency [Hz] time [s] log amplitude frequency

  33. 4 frequency [kHz] 0 0 time [s] 6 /e:/ /a;/ /i:/ /o:/ /u:/

  34. Speech production

  35. /j/ /u/ /ar/ /j/ /o/ /j/ /o/

  36. 5 Fourier transform of the signal s(m) multiplied by the window w(n-m) Spectrum is the line spectrum of the signal convolved with the spectrum of the window Spectral resolution of the short-term Fourier analysis is the same at all frequencies. frequency [kHz] 0 1.2 0 time [s]

  37. W(m) Short-term discrete Fourier transform

  38. Homework

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