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  • based on the frequencies that filters attenuate or boost, they maybe classified into different categories
  • low pass filter:
    • lets through only low frequencies
    • kills high frequencies
  • high pass filter:
    • reverse operation of low pass filter
  • bandpass filter:
    • allow a middle band
    • kills low end and high end
  • ideal filers:
    • theoretical best performance filters in each class of filters
    • similar to an ideal engine
    • cannot be implemented in real life
    • a useful paradigm to understand limitations of real-world filters

frequency classification

  • based on the shape of magnitude response of filters, they can be categorized into four types
    • lowpass
      • let low frequencies live and kill everything else
    • highpass
      • let high frequencies live and kill the rest
    • bandpass
      • let a band of central frequencies through and kill all else
    • allpass
      • let all frequencies through
      • the magnitude curve is a constant through all frequencies
  • this mirrors the time-domain filter classification

  • moving-average and leaky-integrator are lowpass filters

  • filters can also be classified based on phase change characteristic
    • linear phase
    • non-linear phase

ideal low pass filter

lp-fil-00

fig: ideal lowpass filter magnitude spectrum

  • \(\omega_c\): cutoff frequency
    • frequencies above \(\omega_c\) are killed
    • below it are let through filter
    • the magnitude response transistions from \(1\) to \(0\) at \(\omega_c\)
    • filter bandwidth: \(\omega_b = 2 \omega_c \)
ideally:
  • lowpass filters are those which let all low band frequencies through
    • low frequency signals are untouched
    • completely attenuates high frequencies
  • magnitude of spectrum is
    • \(1\) for the low frequency pass band
    • \(0\) for the high frequency stop band
  • for this, magnitude response should be a real function
    • zero phase filter
    • no delay is added by the filter

formal low pass filter

\[ \begin{align} H(e^{j \omega} ) & = \Bigg \{ \begin{matrix} 1 & \text{ for } \vert \omega \vert \leq \omega_c \\ 0 & \text{ otherwise } \end{matrix} & (2\pi\text{-periodicity implicit})\
\end{align} \]

  • perfectly flat passband
  • infinite attenuation in the stopband
  • zero-phase (no-delay)
impulse response of ideal lowpass filter

\[ \begin{align} h[n] & = IDFT \{ H(e^{j\omega}) \} \
& = \frac{1}{2\pi} \int_{-\pi}^{\pi} H( e^{j\omega} ) e^{j \omega n} d \omega \
& = \frac{1}{2\pi} \int_{-\omega_c}^{\omega_c} e^{j \omega n} d \omega \
& = \frac{1}{\pi n} \frac{e^{j \omega_c n} - e^{-j \omega_c n}}{2j} \
& = \frac{\sin \omega_c n}{\pi n} \
\end{align} \]

lp-fil-01

fig: ideal lowpass filter impulse response

  • response has a nice oscillatory shape
  • response is an infinite support impulse response
    • infinite both to the right and left
  • no matter how the convolution is computed, there will always be an infinite number of operations to compute
  • this is the ideal behavior and causes issue in real world filter implementation
    • cannot compute the output in a finite amount of time
  • this behavior is approximated to build filters that respond in finite time
    • approximation for computable, usable, real-world filters
  • the impulse response decays very slowly over time; i.e. @ rate \(\frac{1}{n}\)
    • a lot of samples are needed for a good approximation

dedicated filter response functions

  • the sinc-rect pair: \[ \begin{align} rect(x) & = \Bigg \{ \begin{matrix} 1 & \vert x \vert \leq \frac{1}{2} \\ 0 & \vert x \vert > \frac{1}{2} \end{matrix} \
    \
    sinc(x) & = \Bigg \{ \begin{matrix} \frac{\sin (\pi x)}{\pi x} & x \neq 0 \\ 1 & x = 0 \end{matrix} \
    sinc(x) & = 0 \text{ when } x \text{ is a non-zero integer } \
    \text{ sinc } & \text{ function here is normalized } \
    \end{align}
    \]
frequency response of lowpass filter
  • frequency response in terms of a \( rect \) function of ideal lowpass filter is \[ rect(\frac{\omega}{2\omega_c}) \]
    • \( \omega_c \): cutoff frequency of lowpass filter
impulse response of lowpass filter
  • impulse response in terms of a \( sinc \) function of ideal lowpass filter is \[ \frac{\omega_c}{\pi} sinc(\frac{\omega_c}{\pi}n) \]
relationship between lowpass filter impulse and frequency response

lp-fil-02

fig: ideal lowpass filter frequency (top) and impulse response (bottom) relationship

  • here:
    • \( \omega_c = \frac{\pi}{3} \)
quirks of the ideal lowpass filter
  • the \( sinc \) function is not absolutely summable
    • consequently, the ideal lowpass is not BIBO stable
  • example: \[ \begin{align} \omega_c & = \frac{\pi}{3} \
    h[n] & = \bigg ( \frac{1}{3} \bigg ) sinc \bigg (\frac{n}{3} \bigg ) \end{align} \]

  • consider a bounded input signal for ideal filter \[ x[n] = sign \bigg \{ sinc \bigg ( -\frac{n}{3} \bigg ) \bigg \} \]

  • to compute output of ideal filter, convolve this input and impulse response of ideal lowpass filter \[ \begin{align} y[0] & = (x*h)[0] \
    & = \frac{1}{3} \sum_{k=-\infty}^{\infty} \Bigg \vert sinc \bigg ( \frac{k}{3} \bigg ) \Bigg \vert \
    & = \infty \end{align}
    \]
    • the convolution is divergent computation

  • however, the divergence is fairly slow lp-fil-03

    fig: slow divergence of the ideal lowpass filter convolution with input \(x[n]\)

derived ideal filters

  • a series of other ideal filters can be derived from the ideal lowpass filter

ideal highpass filter

de-fil-00

fig: ideal highpass filter magnitude spectrum

formal ideal highpass filter

\[ \begin{align} H_{hp}(e^{j\omega}) & = \Bigg \{ \begin{matrix} 1 & \text{ for } \pi \leq \vert \omega \vert \leq \omega_c \\ 0 & \text{ otherwise } \end{matrix} & (2\pi\text{-periodicity implicit}) \
\
H_{hp} (e^{j\omega}) & = 1 - H_{lp} (e^{j\omega}) & \text{(highpass-lowpass relationship in frequency)} \
\
h_{hp}[n] & = \delta[n] - \frac{\omega_c}{\pi}sinc\bigg ( \frac{\omega_c}{\pi} \bigg ) & \text{(highpass-lowpass relationship in time)} \
\end{align} \]

  • it can be seen that the ideal highpass is a complementary filter of an ideal lowpass filter in the frequency domain

ideal bandpass filter

de-fil-01

fig: ideal bandpass filter magnitude spectrum

  • this is derived from the ideal lowpass filter by modulating them with a cosine wave

de-fil-02

fig: ideal lowpass filter modulated with cosine wave to obtain ideal bandpass

formal ideal bandpass filter

\[ \begin{align} H_{bp}(e^{j\omega}) & = \Bigg \{ \begin{matrix} 1 & \text{ for } \vert \omega \pm \omega_0 \vert \leq \omega_c \\ 0 & \text{ otherwise } \end{matrix} & (2\pi\text{-periodicity implicit}) \
\
h_{bp}[n] & = 2 \cos(\omega_0 n) \frac{\omega_c}{\pi} sinc \bigg ( \frac{\omega_c}{\pi} n \bigg ) & \text{(highpass-lowpass relationship in time)} \
\end{align} \]

demodulation - frequency domain

  • time domain demodulation concepts
    • apply sinusoidal modulation to \(x[n]\) \[ y[n] = x[n] \cos \omega_0 n \]
    • demodulate by multiplication with carrier \[ x\prime[n] = y[n]\cos \omega_0 n \]
    • demodulated signal contains unwanted high frequency components
    • these unwanted high frequency components are filtered out with a lowpass filter

filtering a demodulated signal

  • consider a signal in the frequency domain \( X(e^{j \omega} ) \)

    demod-fd-00

    fig: signal in frequency domain \( X(e^{j \omega}) \)

  • this is then modulated to get two half amplitudes at the modulation frequency

    demod-fd-01

    fig: modulate \( X(e^{j \omega}) \) signal in frequency domain

  • this has a periodicity of \(2\pi\) in the frequency domain

    demod-fd-02

    fig: extended bounds to reveal periodicity of modulated signal \( X(e^{j \omega}) \)

  • this is then multiplied by \(\cos\omega_0 \) to demodulate, which yields two components

    demod-fd-03

    fig: one copy of the demodulated signal

    demod-fd-04

    fig: second copy of the demodulated signal

  • to get the full demodulated signal, the two components are summed together to get

    demod-fd-05

    fig: sum of the components of the demodulated signal

  • examining a single period of this summation in \( [-\pi,\pi] \)

    demod-fd-06

    fig: single period (between \( [-\pi,\pi] \)) of the summed signal

  • here, spurious high frequency components near \(\pi \) has to be filtered out

  • this is achieved using a lowpass filter

    demod-fd-07

    fig: low pass filter applied to demodulated signal

  • the original signal is obtained as the output of the filter

    demod-fd-08

    fig: original signal obtained from filtering demodulated signal

references