the values h[m] are the kernel these are the weighting factors in the calculation h[m] represents the time domain impulse response DFT of h[m] gives the transfer function H(ω) (frequency domain) in the previous lecture, we looked at the moving average filter let’s look at some more examples…
Exponential Kernel
h[m] values are a decaying exponential
i.e. earlier values of x[n] are weighted less
write exponential kernel as continuous function: h(t) = e−bt u(t)
Fourier transform:
1-pole low pass transfer function
Gaussian Kernel
h[m] are filled with values from a Gaussian function Fourier transform of a Gaussian is … a Gaussian
transfer function is a low-pass filter
Gaussian kernels used in 2D image processing
Gaussian blur to smooth out noise also used for highpass sharpening
How to Make a High-Pass Filter?
technique is called “spectral inversion”
High-Pass Filter Explained
a delta function kernel: y[n] = x[n], is “all-pass” basically, you take what would be the output from the low pass filter and subtract it off from the original
remember: the digital filter acts in time domain
High Pass Math !"#$%&'((%)'*$ here's here's the the math... math... time time domain domain
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how it's done: take the low-pass digital filter kernel, how it's done: take the low-pass digital filter kernel, change the sign of all the kernel samples, then add one change thefunction) sign of all then t=0 add one (the delta at the the kernel centre samples, of symmetry, (the delta function) at the centre of symmetry, t=0
Sinc Kernel
an ideal low-pass filter has a square frequency response flat passband (e.g. unity gain), infinitely sharp cutoff, zero stopband
inverse Fourier transform is the sinc function in time domain
Sinc Kernel Filter Features
sinc(t) is the impulse response → the kernel in theory
the sinc function extends to infinity, both + and − the sinc function is non-causal, has values for t0
in a real implementation
non-causal is not a big deal
the kernel must be chopped so it has finite length
can add a time shift the Fourier transform of a time shift is a phase shift (and it’s a “linear” phase shift which we know is good for filters) shift the kernel h[m] to get a causal filter again so that h[m] = 0 for m < 0 mathematical a “window” function multiplies the sinc kernel
otherwise, the sinc kernel would be an ideal filter
Truncated Sinc – Non Ideal
truncate the kernel to M+1 points, where M is even
shift so that there are no t < 0 values what's the effect of the shift?
it's just a time delay between input and output; still linear phase so this has no impact on pulse shape
Window Function multiple by a window function to smooth the abrupt ends improves passband ripple and stopband attenuation slower roll-off
Window Function
non-ideal frequency domain response is due to abrupt truncation (finite length)
making the truncation less abrupt… do this by multiplying the truncated sinc function with a smoothly tapered curve at the ends → improves the frequency response
Blackman or Hamming Window
Blackman window
Hamming window
window functions go from 0 to M also
Filter Performance
Hamming windowed sinc has ~20% faster roll-off than Blackman windowed sinc
Blackman has -74 dB stopband attenuation while Hamming has only -53 dB
Blackman passband ripple 0.02% while Hamming is 0.2%
in units of sampling frequency e.g. M = 40, width = 0.1 fs
trade-off is between computation time vs. sharpness of filter
Windowed Sinc Filter Design cont'd
note: roll-off sharpness does not change with cutoff frequency final expression:
fc is a fraction of sampling rate between 0 and 0.5 M is an even integer i runs from 0 to M K is a normalization constant (sets unity gain for DC) for i = M/2 (at the sinc peak), h[i] = 2 π fc K
sinc
Blackman
Max'ed Out Windowed Sinc!
how to get -148 dB stopband attenuation?
two-stage Blackman windowed sinc filter
how to get incredible roll-off?
32,001-point Blackman (from pass to stop in 0.000125 fs)
e.g. fs=8 kHz; transition width is 1 Hz
0.02% ripple and residue
Digital Filters Summary straightforward to design straightforward to use near optimum frequency domain performance achievable linear phase → good pulse shape fidelity in time domain also only thing you trade off for higher performance is computation time (due to longer kernel)
Band Pass
cascading filters works; also equivalent to convolution
