contents

  • ideal filters cannot be implemented in real life
  • constraints for a real filter:
    • linearity
    • time-invariance
    • realizability

CCDE


CCDE represent causal LTI systems


  • CCDE: Constant Coefficient Differential Equation
    • a linear combination of past input and output values
    • scaled by constants

[ \sum_{k=0}^{N-1} a_k y[n-k] = \sum_{k=0}^{M-1} b_k x[n-k] ]

  • uses (M) inputs and (N) outputs
    • MIMO: Multiple Input Multiple Output systems
  • LTI systems can only contain
    • additions
    • delays
    • multiplications by constant (scaling)

dsp-operator-filter-reqs

fig: SISO LTI filter requirements

  • a causal LTI system’s output is a linear function of past values of the input and output

CCDE frequency response

  • (z)-transform is a tool to find the frequency response of the system equation
    • is a power series
    • maps a discrete time sequence (x[n]) to a function of the complex variable (z)
  • formally, the (z)-transform is given by \[ X(z) = \sum_{n = -\infty}^{\infty} x[n]z^{-n}, \text{ } z \in \mathbb{C} ]
  • (z)-transform is a generalization of the DTFT for points in the whole complex plane - the (z)-transform of (x[n]) at (z = e^{j\omega}) is the same as the DTFT of (x[n]) \[ X(z)\vert_{z=e^{jw}} = DTFT { x[n] } ]

properties of the (z)-transform

  • linearity \[ \mathcal{Z}{ \alpha x[n] + \beta y[n] } = \alpha X(z) + \beta Y(z) ]
  • time-shift \[ \mathcal{Z} { x[n-N] } = z^{-N} X(z) ]

application to CCDE

  • using the linearity and time-shift properties, the following relationship is obtained

[ \begin{align} \sum_{k=0}^{N-1} a_k y[n-k] & = \sum_{k=0}^{M-1} b_k x[n-k] & \rightarrow \text{CCDE} _
_
_Y(z) \sum{k=0}^{N-1} a_k z^{-k} & = X(z) \sum_{k=0}^{M-1} b_k z^{-k} & \rightarrow z\text{-transform}

Y(z) & = H(z)X(z)
\end{align} ]

  • here, the (z)-transform of the output is the product of the (z)-transform of the input with a function (H(z)) [ H(z) = \frac{\sum_{k=0}^{M-1} b_k z^{-k}}{\sum_{k=0}^{N-1} a_k z^{-k}} ]
  • (H(z)) is the ratio of two polynomials in variable (z)
  • the coefficients of the polynomials are the coefficients that appear in the CCDE
    • in the numerator: coefficients of CCDE input (b_k)
    • in the denominator: coefficients of CCDE output (a_k)
  • to get frequency response of system defined by CCDE, set (z = e^{j\omega}) [ \begin{align} H(z) & = H(e^{j\omega})
    Y(e^{j\omega}) & = H(e^{j\omega}) X(e^{j\omega}) \end{align} ]
  • this is in the form of the relationship between the input and output of a filtering operation
    • so (H) is a filter
    • (H)’s frequency response is computed by the polynomial ratio obtained by the frequency response

existence and convergence

  • each case the (z)-transform is applied to has a Region Of Convergence (ROC)
    • ROC determines points on the complex plane where the (z)-transform exists
  • existence of the (z)-transform is the same as the convergence of the power series that defines the transform
    • ROC is where the power series can converge
  • since (z)-transform is a power series, when it converges, it converges absolutely
    • i.e. convergence depends only on the magnitude of (z), not on its phase
  • absolute convergence condition \[ \vert X(z) \vert < \infty \Leftrightarrow \sum_{n = -\infty}^{\infty} \vert x[n] z^{-n} \vert < \infty ]

three cases of ROC

  • (z)-transform of finite-support sequence
    • the transform is the sum of a finite number of terms
    • so convergence is guaranteed
  • in other cases, the ROC of the (z)-transform has circular symmetry
    • depends only only on (\vert z \vert)
    • if it converges on one point of the complex plane, it converges on all points with the same magnitude
      • this is a circle on the complex plane
  • ROC for causal sequences
    • extends from a circle to infinity
    • ROC extends outside a circle on the complex plane,
    • not inside the circle

rational transfer function

  • in the (z) domain, a CCDE is represented as (H(z)), which is the ratio of tw polynomials of (z^{-1})
    • this polynomial is called a rational transfer function
  • an LTI transfer function: [ H(z) = \frac{ b_0 + b_1 z^{-1} + \ldots + b_{M-1} z^{M-1} }{a_0 + a_1 z^{-1} + \ldots + a_{N-1} z^{N-1}} ]
    • ratio of polynomials
    • polynomial of degree (M-1) in numerator
    • polynomial of degree (N-1) in denominator
  • the frequency response of a filter is equal to this transfer function evaluated at (z = e^{j\omega} )
  • transfer function can also be expressed as [ H(z) = C \frac{\prod_{n=1}^{M-1} (1 - z_n z^{-1}) }{\prod_{n=1}^{N-1} (1 - p_n z^{-1}) } ]
    • ( z_n ): zeros of the transfer function
      • roots of the first order terms in the numerator
      • they send the transfer function to zero
      • represented by filled up ‘O’ in the complex plane
    • ( p_n ): poles of the transfer function
      • roots of the first order terms in the denominator
      • send the denominator to zero
      • so they are trouble spots for ROC
      • represented by ‘X’ in the complex plane
  • this transfer function is analyzed for the ROC of the system defined by the transfer function

causal system ROC

  • for a causal system, we know that the ROC
    • extends outwards from a circle
    • cannot include poles
    • zeros do not affect the region of convergence
  • so ROC extends outwards from a circle touching the largest-magnitude poles
    • this circle size is determined by the poles
    • the circle can touch the pole but not have it in the ROC

zeros-and-poles

fig: plot of zeros and poles of a causal system in the complex plane

zeros-and-poles-ROC

fig: circle and shaded area indicates ROC for a causal system

stability criterion

  • the ROC definition obtained from the poles of transfer function helps defines the stability of the system
  • a stability criterion which depends on knowing the transfer function of the system can be defined
  • the necessary and sufficient condition for BIBO stability is impulse response must be absolutely summable \[ \sum_{n=-\infty}{\infty} \vert h[n] \vert < \infty ]
  • absolute convergence property of (z)-transform
    • when (z)-transform exists for (\vert z \vert = 1)
    • (z)-transform converges absolutely
    • this implies that the impulse response is absolutely summable
  • conversely, if impulse response is absolutely summable
    • it is implied that (z)-transform converges absolutely, and
    • (z)-transform exists for (\vert z \vert = 1)
  • so, the stability criterion for a system is
    • the ROC includes the unit circle
“system is stable only if its ROC contains the unit circle”
  • this means that all the poles of the transfer function must be inside the unit circle

zeros-and-poles-ROC-unit-circle-stable

fig: stable system - poles inside unit circle, ROC contains the unit circle

zeros-and-poles-ROC-unit-circle-unstable

fig: unstable system - poles outside unit circle, unit circle not in ROC

circus-tent method

  • this is a method to estimate frequency response of a system from the complex plane pole-zero plot

procedure

  • assume that the magnitude of the (z)-transform is like a rubber sheet over the complex plane
  • this sheet glued to the ground at the zeros of the transfer function
  • the poles push the rubber sheet up
  • frequency response in magnitude is the profile of the sheet around the unit circle

example

  • consider the following pole-zero plot
    • has two zeros and two poles
    • complex poles always exist in pairs as complex conjugates
      • this true for all real valid systems

circus-tent-method

fig: circus tent method - stable system ROC, includes unit circle

  • consider the complex plane in a 3D perspective
    • the horizontal plane in the complex plane
    • the height is the magnitude of the frequency response (H(z))of the transfer function
    • assume ( H(z) = 1 ) as the starting point for the rubber sheet

circus-tent-method

fig: circus tent method - 3D perspective of frequency response of transfer function

  • estimation process:
    • glue the rubber sheet to the complex plane where the zeros are fig: circus tent method - frequency response glued to ground at zeros
    • poles at poles push the sheet to infinity fig: circus tent method - poles pushing the frequency response fabric
    • this is the shape of the rubber sheet having considered all four components on the complex plane
    • the fourier transform is the value of the (z)-transform around the unit circle
      • this is the level curve of the rubber sheet computed along the unit circle
        fig: circus tent method - level curve on fabric along unit circle
    • plotting this level circle in ( [-\pi,\pi] ) fig: circus tent method - frequency response obtained by spreading out level curve in ( [-\pi,\pi] )
  • the circus-tent method provides understanding about the relationship between
    • the frequency response and
    • the zero-pole plot obtained from the system transfer function

references