Q4Digital Signal Processing

(a) H(z), Poles and Zeros for y(n)+3/4y(n-1)+1/8y(n-2)=x(n)+x(n-1)

Taking the z-transform of both sides of the difference equation (using the time-shift property, Z{y(n-k)}=z^-k*Y(z)):

Zero: setting the numerator to zero, 1+z^-1=0, gives z^-1=-1, i.e., z=-1. There is a single zero at z=-1.

Poles: setting the denominator to zero and multiplying through by z^2 to clear the negative powers: z^2 + (3/4)z + (1/8) = 0. Using the quadratic formula:

giving z1 = (-0.75+0.25)/2 = -0.25 and z2 = (-0.75-0.25)/2 = -0.5. Both poles lie inside the unit circle (|z|<1 for both -0.25 and -0.5), confirming the system is stable.

(b) Magnitude Response for y(n)+1/2*y(n-1)=x(n)-x(n-1)

Taking the z-transform: Y(z)[1+(1/2)z^-1] = X(z)[1-z^-1], giving:

Substituting z=e^jw to evaluate the frequency response:

The magnitude is computed as the ratio of the numerator's magnitude to the denominator's magnitude, each evaluated using the standard identity |1+re^(-jw)| = sqrt(1+r^2+2r*cos(w)):

Evaluating at sample frequencies:

• At w=0: numerator=sqrt(2-2)=0, so |H(e^j0)|=0 - the system completely blocks DC (zero frequency), consistent with the zero at z=1 (since z=e^j0=1 corresponds exactly to the zero location of the (1-z^-1) numerator factor).
• At w=pi/4: numerator=sqrt(2-2*0.7071)=sqrt(0.5858)=0.7654, denominator=sqrt(1.25+0.7071)=sqrt(1.9571)=1.3990, giving |H|=0.7654/1.3990=0.5471.
• At w=pi/2: numerator=sqrt(2-0)=1.4142, denominator=sqrt(1.25+0)=1.1180, giving |H|=1.4142/1.1180=1.2649.
• At w=pi: numerator=sqrt(2+2)=2.0, denominator=sqrt(1.25-1)=sqrt(0.25)=0.5, giving |H|=2.0/0.5=4.0.

Result and interpretation: the computed magnitude response values (0, 0.5471, 1.2649, 4.0 at w=0, pi/4, pi/2, pi respectively) show a monotonically increasing magnitude response from a null at DC to a maximum at w=pi (the highest representable digital frequency), confirming that this system exhibits high-pass filter characteristics - it strongly attenuates low-frequency (near-DC) signal content while passing and even amplifying (gain greater than 1) high-frequency content near w=pi, a direct consequence of the difference-operation-like numerator (1-z^-1), which is the standard first-difference/high-pass building block widely used in simple digital high-pass filter and edge/transient-detection applications.

The pole locations obtained for the first system, z=-0.5 and z=-0.25 (both real, negative, and strictly inside the unit circle), indicate a stable causal system whose impulse response consists of a sum of two decaying, alternating-sign exponential (geometric) sequences - the presence of two negative real poles causes the impulse response and step response to exhibit a characteristic damped oscillatory (alternating) transient behavior at the sample-to-sample level even though there is no complex-conjugate pole pair present, a subtlety that is sometimes overlooked but is a direct consequence of any negative real pole contributing a term of the form (-r)^n = r^n cos(pin) to the time-domain response, which alternates sign every sample regardless of whether the pole is real or complex.

The zero at z=-1, located precisely on the unit circle at the Nyquist frequency (w=pi), causes the system's magnitude response to vanish exactly at that discrete frequency, |H(e^jpi)|=0, meaning the system completely blocks any input signal component oscillating at exactly the Nyquist rate (alternating sign every sample) - this specific pole-zero configuration is characteristic of a low-pass-like digital filter section that passes low-frequency (slowly varying) signal content while progressively attenuating higher-frequency content, reaching complete rejection precisely at the highest representable digital frequency.

For the second system, y(n)+(1/2)y(n-1)=x(n)-x(n-1), the computed magnitude-response values at the four standard reference frequencies (0, pi/4, pi/2, and pi) trace out a clearly monotonically increasing magnitude-response curve from a null at DC (w=0, where the numerator's (1-e^-jw) factor vanishes identically) up to a maximum value at the Nyquist frequency (w=pi) - this behavior identifies the system unambiguously as a high-pass-type digital filter, a classification that follows directly and consistently from its transfer function structure, since the numerator's zero at z=1 (corresponding to w=0) always suppresses any DC or slowly varying signal component, while the single pole at z=-0.5 contributes a moderate resonant boost toward the higher end of the digital frequency range, reinforcing the overall high-pass shape of the magnitude response.

It is instructive to note the general design principle these two example systems both illustrate: for simple first- and second-order digital IIR systems, the qualitative frequency-selectivity behavior (low-pass, high-pass, band-pass, or notch) can very often be determined directly by inspection of zero locations relative to the unit circle (a zero placed at or near a particular frequency point on the unit circle strongly suppresses the response at that frequency) combined with pole locations relative to the unit circle (a pole placed near a particular frequency point on the unit circle strongly boosts, or resonates, the response near that frequency) - this pole-zero-placement intuition is the conceptual foundation underlying more systematic digital filter design techniques, including the frequency-transformation methods (impulse invariance, bilinear transformation) used to design higher-order filters elsewhere in this examination from classical analog prototypes.

It is worth noting that these two example systems, one exhibiting low-pass character (the first, with its unit-circle zero at the Nyquist frequency) and one exhibiting high-pass character (the second, with its unit-circle zero at DC), together illustrate the complementary roles that pole and zero placement play in shaping a digital filter's frequency selectivity, and this pole-zero-placement intuition extends naturally to the design of simple notch or resonant filters by combining a zero placed precisely at an unwanted frequency (to null it out completely) with a nearby pole placed at a slightly smaller radius (to sharpen the notch's rejection bandwidth), a technique commonly used for removing narrowband interference, such as 50Hz or 60Hz power-line hum, from a wideband signal without requiring a high-order general-purpose band-stop filter design.

A further useful observation is that both example systems in this question are first- or second-order IIR sections of exactly the type that would appear as building blocks within a larger cascade or parallel realization of a higher-order digital filter, meaning the pole-zero analysis and magnitude-response evaluation techniques demonstrated here for these individual low-order sections are directly applicable, section by section, to analyzing the behavior of much more complex, higher-order digital filters designed and realized elsewhere in this examination paper.

In summary, this pair of examples demonstrates the standard workflow for characterizing a digital IIR system from its difference equation: deriving H(z), locating its poles and zeros, and evaluating the magnitude response at representative frequencies to classify the filter's overall frequency-selectivity behavior.

This pole-zero-based classification approach, requiring only inspection of the transfer function's numerator and denominator roots relative to the unit circle, is generally far faster in an examination setting than plotting the full magnitude response numerically, and is therefore the recommended first step whenever a quick, qualitative frequency-selectivity classification of a low-order digital system is required.