# 1 1. The ROC of the z-transform H(z) of the impulse response sequence h[n] is defined

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1 A causal LTI digital filter is BIBO stable if and only if its impulse response h[ is absolutely summable, i.e., S h [ < n We now develop a stability condition in terms of the pole locations of the transfer function H(z) The ROC of the z-transform H(z) of the impulse response sequence h[ is defined n by values of z r for which h[ r is absolutely summable Thus, if the ROC includes the unit circle z, then the digital filter is stable, and vice versa In addition, for a stable and causal digital filter for which h[ is a right-sided sequence, the ROC will include the unit circle and entire z-plane including the point z An FIR digital filter with bounded impulse response is always stable On the other hand, an IIR filter may be unstable if not designed properly In addition, an originally stable IIR filter characterized by infinite precision coefficients may become unstable when coefficients get quantized due to implementation 3 4 Example - Consider the causal IIR transfer function. 845z z Amplitude 6 4 h[ 5 The plot of the impulse response coefficients is shown on the next slide Time index n As can be seen from the above plot, the impulse response coefficient h[ decays rapidly to zero value as n increases

2 7 The absolute summability condition ofh[ is satisfied Hence, H(z) is a stable transfer function Now, consider the case when the transfer function coefficients are rounded to values with digits after the decimal point: ^. 85z +. 85z 8 ^ A plot of the impulse response of h[ shown below Amplitude 6 4 h^[ Time index n is 9 In this case, the impulse response coefficient h^[ increases rapidly to a constant value as n increases Hence, the absolute summability condition of is violated Thus, H^ (z) is an unstable transfer function The stability testing of a IIR transfer function is therefore an important problem In most cases it is difficult to compute the infinite sum S n h [ < For a causal IIR transfer function, the sums can be computed approximately as S K K n h [ The partial sum is computed for increasing values of K until the difference between a series of consecutive values of S K is smaller than some arbitrarily chosen small number, which is typically 6 For a transfer function of very high order this approach may not be satisfactory An alternate, easy-to-test, stability condition is developed next Consider the causal IIR digital filter with a rational transfer function H(z) given by M p k k k z N d k k k z Its impulse response {h[} is a right-sided sequence The ROC of H(z) is exterior to a circle going through the pole furthest fromz

3 3 But stability requires that {h[} be absolutely summable This in turn implies that the DTFT H ( e ) of {h[} exists Now, if the ROC of the z-transform H(z) includes the unit circle, then j ω ) H ( e z e 4 Conclusion: All poles of a causal stable transfer function H(z) must be strictly inside the unit circle The stability region (shown shaded) in the z-plane is shown below unit circle j Im z j j stability region Re z 5 Example - The factored form of is.845z z (.9z )(.943z ) which has a real pole at z.9 and a real pole at z.943 Since both poles are inside the unit circle, H(z) is BIBO stable 6 Example - The factored form of is H ^ ( z).85z +. 85z H^ ( z) ( z )(.85z ) which has a real pole on the unit circle at z and the other pole inside the unit circle Since both poles are not inside the unit circle, H(z) is unstable Types of Types of 7 The time-domain classification of an LTI digital transfer function sequence is based on the length of its impulse response: - Finite impulse response (FIR) transfer function - Infinite impulse response (IIR) transfer function 8 Several other classifications are also used In the case of digital transfer functions with frequency-selective frequency responses, one classification is based on the shape of the magnitude function H ( e ) or the form of the phase function θ(ω) Based on this four types of ideal filters are usually defined 3

4 A digital filter designed to pass signal components of certain frequencies without distortion should have a frequency response equal to one at these frequencies, and should have a frequency response equal to zero at all other frequencies The range of frequencies where the frequency response takes the value of one is called the passband The range of frequencies where the frequency response takes the value of zero is called the stopband 9 Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below: Lowpass Highpass Bandpass Bandstop Lowpass filter: Passband - ω ω c Stopband - ω c < ω π Highpass filter: Passband - ω c ω π Stopband - ω< ω c Bandpass filter: Passband - ωc ω ωc Stopband - ω< ωcand ω c < ω π Bandstop filter: Stopband - ωc < ω< ωc Passband - ω ωc and ω c ω π 3 The frequencies ωc, ωc, and ω c are called the cutoff frequencies An ideal filter has a magnitude response equal to one in the passband and zero in the stopband, and has a zero phase everywhere 4 Earlier in the course we derived the inverse DTFT of the frequency response H ( j ω LP e ) of the ideal lowpass filter: sinω n h n c LP [ ], < n < πn We have also shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable 4

5 5 Also, h LP [ is not causal and is of doubly infinite length The remaining three ideal filters are also characterized by doubly infinite, noncausal impulse responses and are not absolutely summable Thus, the ideal filters with the ideal brick wall frequency responses cannot be realized with finite dimensional LTI filter 6 To develop stable and realizable transfer functions, the ideal frequency response specifications are relaxed by including a transition band between the passband and the stopband This permits the magnitude response to decay slowly from its maximum value in the passband to the zero value in the stopband Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband Typical magnitude response specifications of a lowpass filter are shown below A second classification of a transfer function is with respect to its phase characteristics In many applications, it is necessary that the digital filter designed does not distort the phase of the input signal components with frequencies in the passband 7 8 One way to avoid any phase distortion is to make the frequency response of the filter real and nonnegative, i.e., to design the filter with a zero phase characteristic However, it is possible to design a causal digital filter with a zero phase For non-real-time processing of real-valued input signals of finite length, zero-phase filtering can be very simply implemented by relaxing the causality requirement One zero -phase filtering scheme is sketched below x[ H(z) v[ u[ H(z) w[ 9 3 u[ v[, y[ w[ 5

6 3 It is easy to verify the above scheme in the frequency domain Let X ( e ), ( e j ω V ), U ( e ), W ( e j ω), and Y ( e ) denote the DTFTs ofx[, v[, u[, w[, and y[, respectively From the figure shown earlier and making use of the symmetry relations we arrive at the relations between various DTFTs as given on the next slide 3 x[ H(z) v[ u[ H(z) w[ ( j ω ( e j ω ) V*( e ), u[ v[, y[ w[ ( j ω ( e j ω ) W*( e j ω V e ) H ( e ) X ( e ), W e ) H ( e ) U ( e ) U Y ) Combining the above equations we get Y( e ) W*( e ) H * ( e ) U *( e ) H ( e ) V ( e ) H* ( e ) H( e ) X( e H ( e ) X ( e ) * ) 33 The functionfftfilt implements the above zero -phase filtering scheme In the case of a causal transfer function with a nonzero phase response, the phase distortion can be avoided by ensuring that the transfer function has a unity magnitude and a linear-phase characteristic in the frequency band of interest 34 The most general type of a filter with a linear phase has a frequency response given by ω D H ( e j ) e which has a linear phase fromω to ω π Note also H ( e ) τ ( ω) D 35 The output y[ of this filter to an input x[ Ae j ωn is then given by D n ( n D) y[ Ae e Ae If x a (t) and y a (t) represent the continuoustime signals whose sampled versions, sampled at t nt, are x[ and y[ given above, then the delay between x a (t) and y a (t) is precisely the group delay of amount D 36 If D is an integer, then y[ is identical to x[, but delayed by D samples If D is not an integer, y[, being delayed by a fractional part, is not identical to x[ In the latter case, the waveform of the underlying continuous-time output is identical to the waveform of the underlying continuous-time input and delayed D units of time 6

7 If it is desired to pass input signal components in a certain frequency range undistorted in both magnitude and phase, then the transfer function should exhibit a unity magnitude response and a linear-phase response in the band of interest Figure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband Since the signal components in the stopband are blocked, the phase response in the stopband can be of any shape Example - Determine the impulse response of an ideal lowpass filter with a linear phase response: H ( LP e ) n e o,, < ω < ωc ω ω π c 4 Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at sinω n n h n c ( o) LP [ ], < n < π( n n ) As before, the above filter is noncausal and of doubly infinite length, and hence, unrealizable o By truncating the impulse response to a finite number of terms, a realizable FIR approximation to the ideal lowpass filter can be developed The truncated approximation may or may not exhibit linear phase, depending on the value of chosen n o If we choose n o N/ with N a positive integer, the truncated and shifted approximation ^ sinωc ( n N /) hlp[, n N π( n N / ) will be a length N+ causal linear-phase FIR filter 4 4 7

8 43 Figure below shows the filter coefficients obtained using the functionsinc for two different values of N Amplitude.6.4. N Time index n Amplitude.6.4. N Time index n 44 Because of the symmetry of the impulse response coefficients as indicated in the two figures, the frequency response of the truncated approximation can be expressed as: N ^ ( ^ ~ H ) [ ] n N / LP e hlp n e e H LP n where H ~ LP (ω), called the zero-phase response or amplitude response, is a real function of ω ( ω) 8

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