Systems (filters)

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

PROCESSING One way of “signal processing”

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

deciBel [dB] Log-log frequency response

Output at time t depends only on the input at time t Memoryless system (amplifier) Frequency response of the system Magnitude (dB) 3 0 phase frequency x

System with a memory (differentiator) Frequency response of the differentiator (high-pass filter) time 0t0t0 in 0t0t0 time out 1 sample delay -

System with a memory (integrator) Frequency response of the integrator (low-pass filter) time 0t0t0 in 0t0t0 time out 1 sample delay +

delay T D - TDTD e.t.c T D =T 1 T D =3T 2 T D =5T 3 Comb filter 1/T D 3/T D 5/T D 1 Frequency response of the system magnitude 0 e.t.c. frequency const

linear system output input nonlinear system output input

noise noisy system

10 ms2 ms Pulse train Its magnitude spectrum

10 ms 2 ms 20 ms

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

time tt 0 frequency  t system Dirac impulseImpulse response time Fourier transform frequency Frequency response Dirac impulse contains all frequencies Fourier transform of the impulse response of a system is its frequency response!

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

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

10 ms2 ms Pulse train Its magnitude spectrum f = 1/ =500 Hz line spectrum with |sinc| envelope 1/t p 2/t p 3/t p frequency 0 continuous |sinc| function tptp ∞ ∞ -

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

 t = ∞  t = 100 ms  t = 13 ms Hz

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

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

Time-Frequency Compromise Fine resolution in one domain (  f-> 0 or  t->0) requires infinite observation interval and therefore pure resolution in the dual domain (  -> or  F-> ) – 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

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

Non-stationary turns into periodic

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

Short-term Discrete Fourier Transform

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

time frequency time

frequency

Analysis window 50 ms time [s]01.2 Analysis window 5 ms time [s]01.2 frequency [kHz] 5 0 frequency log amplitude frequency

frequency [Hz] time [s] frequency log amplitude

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

Speech production

/j/ /u/ /a r / /j/ /o/ /j/ /o/

time [s] frequency [kHz] 5 0 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.

time TT t0t0 fourier transform s(f,t 0 ) spectrum of the short segment time frequency

Short-term discrete Fourier transform W(m)