digital signal processing assignment

In the real world there is a different types of analog signals such as videos, data, voice…etc that the people deal with it daily. On the other hand there is the machines and computers which deal with digital signals to operate, process and control so, the machines to work the analog signal should be convert into discrete or digital form. This operation is called analog to digital convertersion.

Figure1: Analog - digital converter

Processes of Analog to digital conversion

An analog-to-digital converter (ADC) is a device or a system convert the analog signals into digital signals. Where, the computers before handle the data through the communication line it should be in a digital form to be read. As an example for the converting is the sound which picked up by the microphone that converted into a digital signal or other example is the light which entering the digital camera. The analog to digital conversion done in main stages as the following:

1. Sampling

Sampling is the first step in the analog-to-digital conversion. It is defined as reduction or converting from an analog or continues form to a digital discrete form at regular discrete moments of a time. Where sampling is the basic step in analog-to-digital conversion. Where The sampling frequency can be calculated from , where fs; Sampling Frequency , Ts; Sampling Time. In addition, sampling is used to digitalize analog values or data mainly and make it suitable or readable for machines. Sampling process has many advantages as an example data compression it is use in machines to save the memory space, less time consumption and low coast.

Figure2: Signal sample at intervals of time

For performing a proper sampling operation, sampling frequency (fs) must be (twice) the value of (fm) maximum frequency. Mathematically, its , where fs; Sampling Frequency, fm; Maximum Frequency. It’s known as Nyquist Rate equation.

Figure3: proper sampling Diagram

Under sampling

, mean aliasing error where the low frequency components and high frequency components overtop.

Figure4: Under sampling Diagram

2. Quantizing

The quantizing is defined as the process of mapping or rounding off the values to fixed quantization levels. As an example for quantization process is rounding off and truncation. This step it represent the data compression and it has large positive effect on minimizing complexity of big input values. The quantize is the device that do this operation. Quantization effect on peak-to-peak of the input sample values and output is assigned a discrete value.

Figure5: Quantization

3. Encoding

Encoding process is the last step in the analog-to-digital conversion. It is defined as the operation of assigning or giving the input coded data values to multiple quantization levels. This type of operation are using a unique encoding systems is related to give each data a different level compare to the other data forms. It has different pros like storing data and reducing the requirement of the memory.

Figure6: Encoding process

The advantage of Digital processing compared to Analog processing

Digital signals processing has many advantage compared to the analog processing signals as following. Firstly, the main pros of digital signals is the precise of the level signal is not vital this is mean are fairly immune to the imperfections of real electronic systems which tend to spoil analog signals.(wikipedia,nd) Secondly, digital signals can transfer the information with higher noise immunity compare to the analog signals where it is effected by whole level of the noise. Moreover, the cost of the digital circuit components are less cost and cheap. In addition the bandwidth which used is less than the analog signals which give more ability to carry more information and also transmit it to long distance. Also, it is use a high rate transmission with wider broadband width and it is much secure. As well known, it is translate or convert the humans video and audio signals into machine languages. It is allows multidirectional transmission simultaneously. Moreover, many mathematics algorithms equation used in digital processing over the analog processing, which gives the input data given variety of characteristics and options which cannot given in analog processing. Also, digital signal cannot be corrupted by the noise. Finally, in digital processing the signals are more flexible due to the programming where it is easily to programed according to the given requirements, which is difficult to done by the analog signal processing.

Filter with specified bandwidths designed using pole-zero placement method of filter design

pole-zero placement is a method used to design fractional order of the digital filters which is directly in the discrete domain through subsequent parameter optimization and pole-zero placement. Moreover, the zeros and poles are placed on the real axis of the circle to guarantee stability and maximize performance.

Whenever, the bandwidth is specified in the pole-zero placement method of filter design the zeros will be on position of the unit circle and the poles will be at the same angle. The poles are move closer or toward the zeros.

In this case as an example a steps of designed a filter with specified bandwidths using pole-zero placement method will be shown.

a) Parameters given:

Fm = 500 Hz

Noise BW = 5 Hz

b) Design steps:

1. First, determine the analog filter cut-off frequency using nyquist rate formula

FS ≥ 2 Fm

Where, as a solution for given example:

FS ≥ 2 Fm

≥ 1000 Hz

*note: the value of chosen Fs should be four times the value of Fm

Fs = 4 × Fm

= 2000 Hz

2. Draw the Pole – zero map of the system (diagram) which show the location of the zero where it will placed on the edge or the boundary of the unit circle and the poles are moving toward or closely to the zeros and the distance that ( r ) moving is called radius. Moreover, the angle between is ( π/4 ). As the diagram below is show:

Fs/8

r

Fs=2000

Fs/2=1000

3. Moving to the next step is calculate the value of ( r ) radius depend in the parameters given. By using the below formula:

Where, Bw = 2 ( 1 – r )

4. Next, determine the transfer function by the bandwidth formula as given below:

Where, Hz= [ ]

*note: the calculator should be in radian mode to calculate cos()

5. Determine the difference equation by divide both numerator and denominator by Z2 and the a cross multiplication by both side of equation. Then, re-arrange the equation.

6. The finite difference equation of the filter is found by inverting the transfer function of H(z):

7. The last step is draw filter structure depend on the previous steps.

Analog to Digital Transformation of filters using Bilinear Transform Method

The bilinear transformation is one of the methods of converting analog into digital form. It is mathematical mapping of variables. It is also, mean converting the S plane to Z plane. This type of transforming happens by using classical filter design techniques, into their discrete equivalents. The bilinear transformation is a change of variables that is linear in both the numerator and denominator. Moreover, It is also can be used produce or create a piecewise constant magnitude response that approximates the magnitude response of an equivalent analog filter. It does not faithfully reproduce the analog filters phase response, however.

These steps are following for analog to digital transformation of filters using bilinear transform method: as an example:

a) Parameters given:

H(s)=

Frequency required, fd= 500HZ

*note: the value of fd equal to the value of fm

b) Design steps:

1. First, determine the analog filter cut-off frequency using nyquist rate formula

FS ≥ 2 Fm

Where, as a solution for given example:

FS ≥ 2 Fm

≥ 2 × 500 Hz

≥ 1000 Hz

*note: the value of chosen Fs should be four times the value of Fm

Fs = 4 × Fm

= 4 × 500

= 2000 Hz

2. calculate the equivalent analog filter cut-off frequency using pre-warping function:

pre-warping frequency is a technique used to produce a more faithful mapping and also to account for the nonlinearity. pre-warp frequency giving as:

=

= 2πfd

=

*note: the calculator should be in radian mode to calculate tan()

Where, as a solution for given example:

=

= = 1 rad/sec

3. Using analog frequency, find H(S) of the analog filter. Where the rule is and known that

So, substitute in given equation in the question.

Where, as the solution will be as following equation:

H(s)=

4. Next step, convert the analog filter to an equivalent digital filter by applying the bilinear z- transform. This transform is achieved by substitute each s in transform function by giving equation:

To simplify the equation multiply both of the numerator and denominator .

The final form of this step of given example after the whole mathematics substitutions will be as:

= 0.293 ×

5. The finite difference equation of the filter is found by inverting the transfer function of H(z):

Where,

Y(n)= 0.293 x(n) + 0.586 x(n-1) + 0.293 x(n-2) – 0.171 y(n-2)

6. The last step is draw filter structure depend on the previous steps.

Design Concept

To progress a proper solution for the company case a following steps should be following:

a) Selection of the sampling rate ( Fs )

Given:

Fm = 500 Hz ( limited frequency )

Known: FS ≥ 2 Fm (nyquist rate)

≥ 2 × 500 Hz

≥ 1000

Choose Fs 4times the Fm

Fs = 4 × Fm

= 4 × 500

= 2000 Hz

b)

(i) pole – zero diagram(s)

3Fs/44

After designing the system with the specifications given it was noticed many results as following mention. Where, the whole calculations were done properly. Started with found the transfer function equation and difference equation which used for the MATLAB simulation to get the output as denominator and numerator . where, the result was four zero’s and four poles -real and conjugate-  in a details mean there was two original poles with their conjugates, and two zeros with their conjugates. Firstly, Selection of the sampling rate was done and then, the diagram of the pole-zero diagram drawn properly. At the frequency of 150Hz , where the zeros were located at the boundary or edge of the circuit of the Z-plane and the poles were moving toward or close to the zeros (edge of the circuit) where, the radius  setting as (r= 0.992). With angle of (3π/20) started from the center of the circuit to the boundary. Moving to the second interference frequency 120Hz  (noise frequency), where the zeros were located at the boundary or edge of the circuit of the Z-plane and the poles were in the center of the circuit. With angle of (3π/25) started from the centre of the circuit to the boundary . The transfer and difference equations were calculated properly and the inverse of the z transform were done successfully. The whole calculation done were used in the MATLAB program to show the graph as mention previously. The mathematical calculations are matches with the software design in MATLAB program.