Digital Filters MCQ Quiz - Objective Question with Answer for Digital Filters - Download Free PDF

Explanation:

Digital Filters in Signal Processing

Definition: Digital filters are used in digital signal processing (DSP) to manipulate or alter the characteristics of a signal. They perform operations such as removing unwanted noise, extracting useful information, or converting signals from one form to another. The design of digital filters involves specifying a desired magnitude characteristic to achieve particular signal processing goals.

Design Considerations: When designing digital filters, one critical aspect is the duration or the impulse response of the filter. The duration of the impulse response impacts the computational complexity, memory requirements, and the overall performance of the filter. The choice of the filter duration depends on the specific requirements of the application.

Correct Option Analysis:

The correct option is:

Option 1: The shortest duration

This option is correct because, in many cases, digital filters are designed to have the shortest possible duration of the impulse response. This approach is often preferred for several reasons:

• Computational Efficiency: A filter with a shorter impulse response requires fewer computations per input sample. This reduces the processing time and makes real-time processing more feasible, especially in applications with limited computational resources.
• Memory Requirements: Shorter impulse responses require less memory for storage. This is crucial in embedded systems and other environments with constrained memory capacity.
• Reduced Delay: Filters with shorter durations introduce less delay in the processed signal. This is important in applications where low latency is critical, such as in telecommunications and real-time control systems.

To further analyze the other options, we can consider the implications of choosing different durations for the filter's impulse response:

Additional Information

Let's evaluate the other options:

Option 2: Medium duration

Choosing a filter with a medium duration might be a compromise between computational efficiency and performance. However, it may not be optimal for applications requiring high efficiency or low latency. Medium duration filters can still be useful in situations where moderate performance is acceptable, and resource constraints are less stringent.

Option 3: The largest duration

Filters with the largest duration tend to have high computational complexity and significant memory requirements. They can provide better frequency resolution and sharper filter characteristics but at the cost of increased processing time and latency. These filters are less suitable for real-time applications and are more commonly used in offline or batch processing scenarios where performance is prioritized over efficiency.

Option 4: Between medium to the largest duration

This option represents a range of durations that offer a balance between performance and efficiency. Filters in this category might be used in applications where some degree of compromise is acceptable, and resources are moderately constrained. However, they still may not be as efficient as filters with the shortest duration.

Conclusion:

The design of digital filters often involves choosing a filter with the shortest duration to optimize computational efficiency, memory usage, and latency. While filters with longer durations can provide better performance in terms of frequency resolution and sharpness, they come with increased complexity and resource requirements. Therefore, for many practical applications, especially those involving real-time processing, filters with the shortest duration are preferred.

Bilinear transform:

In the context of designing Digital IIR Filters using the Bilinear Transform method, the matching between the design specifications and their characteristics is based on understanding the fixed parameters and their implications in the design process.

• The design procedure has to start with the evaluation of the order of the filter necessary to meet the specifications in terms of the desired attenuation, transition bandwidth and pass-band deviation. (P)
• The filter is completely specified, and the transition bandwidth is directly obtainable during the design procedure. (Q)
• The design is completely determined for the Butterworth filter case by obtaining the value of the attenuation at fa directly. (R)

Now, let's match this:

1. P - 1 (The evaluation of the filter order based on attenuation, transition bandwidth, and pass-band deviation requires fixed values for N (order) and \(\Delta f\) (transition bandwidth).)
2. Q - 2 (When both \(\Delta f\) (transition bandwidth) and (pass-band deviation) are fixed, the filter specifications are completely defined.)
3. R - 3 (For Butterworth filter cases, with fixed values of N (order) and δ (pass-band deviation), the design becomes completely determined by acquiring the attenuation value at f_a.)

Here, option 3 is correct.

Fourth order Butterworth low pass filter is the most complex. 

The correct order of complexity is as follows:

Fourth order Butterworth low pass filter > First order low pass filter > Sallen‐Key filter > Non‐inverting amplifier

As, the order increases, the complexity also increases and the operational amplifier is the most simple one among the following. 

Pole zero plot of band pass filter:

Pole zero plot of band stop/elimination filter:

Pole zero plot for low pass filter:

Pole zero plot of high pass filter:

Image frequency rejection ratio \(\alpha = \sqrt {1 + {Q^2}{\rho ^2}} \)

\(\rho = \frac{{{f_{si}}}}{{{f_s}}} - \frac{{{f_s}}}{{{f_{si}}}}\)

Image frequency: The signal which causes interference is called ‘image frequency’

Image frequency and intermediate frequency related as:

fsi = fs + 2 IF

fs: carrier frequency of the tuned station

fsi: image frequency

IF: Intermediate frequency

The frequency response of the second-order low pass filter is identical to that of the first-order type except that the stop band roll-off will be twice the first-order filters at 40 dB/decade (12 dB/octave). 

The response of the second order filter is:

An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier. Amplifiers included in a filter design can be used to improve the cost, performance and predictability of a filter.

Types of active filters. 

• Butterworth 
• Chebyshev 
• Bessel
• Elliptic filters.

Bessel Filter:

• The Bessel filter provides ideal phase characteristics with an approximately linear phase re­sponse upto nearly cut-off frequency.
• Though it has a very linear phase response but a fairly gentle skirt slope.
• For applications where the phase characteristic is important, the Bessel filter is used.
• It is a minimal phase shift filter even though its cut-off characteristics are not very sharp. Hence it is known as a Constant Phase Delay filter.
• It is well suited for pulse applica­tions.

FIR Filters:
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration because it settles to zero in finite time.

This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying).
The impulse response of an Nth-order discrete-time FIR filter (i.e., with a Kronecker delta impulse input) lasts for N + 1 samples and then settles to zero.
FIR filters can be discrete-time or continuous-time, and digital or analog.

Summary of Windows Design in case of FIR filters:

Bilinear Transformation:

After the frequency scaling and transformation into a desirable type of filter have been performed, it is necessary to transform the resulting analog filter into a digital one.

Analog filter is stable if the poles of the transfer function are located in the left half of s plane, whereas digital filter is stable if the poles are located within the unit circle. 

For this reason, the transformation must provide that the left half of s plane coincides with the area within the unit circle of z plane, as shown:

One of the most commonly used methods of transforming analog filters into appropriate IIR filters is known as bilinear transformation. It is defined via expression:

\(s=\frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}}\)

• In Bilinear Transformation, we carry out conformal mapping in which the jΩ axis is mapped onto the unit circle in the ‘z’ plane. This is a one-to-one mapping that does not bring along with it the issue of aliasing.
• All points in the left-hand side of the s-plane get mapped inside the unit circle in the z-plane. And all points in the right-hand side of the s-plane get mapped outside the circle in the z-plane.
• First, we will transform an analog filter, get H(z), and then get a relationship between s and z.

Properties of FIR filter:

• Impulse Response is of finite duration
• It is a non-recursive system because there is no feedback path present in the FIR filter.
• They are always be designed as a linear phase.
• Stability is guaranteed is the FIR system.
• In FIR, to obtain the same frequency response as IIR large number of additions & multiplication is reviewed, so the speed is very slow.

Properties of IIR filter:

• The impulse response is of infinite duration.
• IIR system is also known as a recursive system because there is a feedback path from output to input.
• IIR system cannot be designed as a linear phase system.
• Stability cannot be guaranteed.
• In the IIR system, fewer multiplications and additions are reviewed, so processing speed is very fast.

Linear phase:

• FIR can be easily designed to have a exact linear phase
• No phase distortion is introduced into the signals to be filtered.
• All frequencies are shifted in time by the same amount.
• Increasing the order of the filter, but keeping all else the same, increases the sharpness of the filter roll-off
• The sharpness of the FIR and IIR filters is very different for the same order.
• Because of the recursive nature of an IIR filter, where part of the filter output is used as an input, IIR filters have sharper roll-off with the same order FIR filter.

The characteristics of analog and digital filters are summarized below:

• The most widely used method of modulation in frequency-division multiplexing is single side band modulation, which in the case of voice signals, requires a bandwidth that is approximately equal to that of the original voice signal. In practice, each voice input is usually assigned a bandwidth of 4KHZ.
• The band to restrict the band of each modulated wave to its prescribed range.
• The resulting band pass filters outputs are next combined in parallel to form the input to the common channel.
• At the receiving terminal, a bank of band-pass filters with their inputs connected in parallel is used to separate the message signals on a frequency occupancy basis.
• Finally the original message signals are recovered by individual demodulators.

Pole zero plot of band pass filter:

Pole zero plot of band stop/elimination filter:

Pole zero plot for low pass filter:

Pole zero plot of high pass filter:

FIR filtering has these advantages over IIR filtering:

1.It can implement linear-phase filtering. This means that the filter has no phase shift across the frequency band. Alternately, the phase can be corrected independently of the amplitude.

2. They are guaranteed to be stable.

3. It can be used to correct frequency-response errors in a loudspeaker to a finer degree of precision than using IIR Filter.

Note:

Explanation:

The transition band characteristics we are usually interested in deal with the shape or steepness of the roll-off between the passband and the stopband. Usually, the shape factor is defined as shown in Figure (which shows an otherwise ideal filter with just the addition of transition bands). It is defined for two levels of attenuation and is usually taken to be the ratio between 3dB of attenuation (passband) and given stopband attenuation.

For example, we could take 3dB to define the passband and 60dB to use the stopband.

Our ideal filter would have a shape factor of unity, but where this is not physically realizable we seek the smallest shape factor we can. For reference, simple RLC filters might have shape factors in the range of ~3, SAW filters in the range of ~1.5.

\(\rm Shape\ Factor=\frac{BW_{60dB}}{BW_{3dB}}\)     ---(1)

Calculation:

Given:

BW at 60 dB = 10 kHz

BW at 3 dB = 4 kHz

From equation (1)

\(\rm Shape \ Factor=\frac{10}{4}=2.5\)

A window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval.

It is normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle.

Hamming window:

The Hamming window is an extension of the Hann window in the sense that it is a raised cosine window of the form

\(h\left( n \right) = \alpha + \left( {1.0 - \alpha } \right)\cos \left[ {\left( {\frac{{2\pi }}{N}} \right)n} \right]\)

with a corresponding spectrum of the form

\(H\left( \theta \right) = \alpha D\left( \omega \right) + \frac{{\left( {1.0 - \alpha } \right)}}{2}\left[ {D\left( {\omega - \frac{{2\pi }}{N}} \right) + D\left( {\omega + \frac{{2\pi }}{N}} \right)\;} \right]\)

The parameter α permits the optimization of the destructive sidelobe cancellation.

The common approximation to this value of α is 0.54, for which the window is called the Hamming window and is of the form

\(H\left( \theta \right) = 0.54 + 0.46\cos \left[ {\left( {\frac{{2\pi }}{N}} \right)n} \right],0 \le n \le N\)