Digital Signal Processing. Lecture Notes. Prof. Dan Cobb. Contents. 1 Introduction. 2. 2 Review of the DT Fourier Transform. 3. Definition and Properties. signal processing operations by changing the program. In analog Several filters need several boards in analog, whereas in digital same DSP processor is. DIGITAL SIGNAL PROCESSING. LECTURE NOTES. nferosexmaufu.gq (III YEAR – II SEM) . (). Prepared by: Mrs. nferosexmaufu.gq, Professor. Department of Electronics.

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Engineering Class handwritten notes, exam notes, previous year questions, PDF free download. Check out the Complete note on Digital Signal Processing Pdf Notes Download. We provide nferosexmaufu.gq Digital Signal Processing study materials to nferosexmaufu.gq student. System. A system is a physical device that performs an operation on a signal. For example, natural signals are generated by a system that responds to a stimulus.

Before we introduce the DFT we consider the sampling of the Fourier transform of an aperiodic discrete-time sequence. Thus we establish the relation between the sampled Fourier transform and the DFT. A discrete time system may be described by the convolution sum, the l Fourier representation and the z transform as seen in the previous chapter. If the signal is or periodic in the time domain DTFS representation can be used, in the frequency domain the spectrum is discrete and periodic. If the signal is non-periodic or of finite duration the frequency domain representation is periodic and continuous this is not convenient to W implement on the computer. Exploiting the periodicity property of DTFS representation the finite duration sequence can also be represented in the frequency domain, which is referred to as Discrete Fourier Transform DFT. DFT is an important mathematical tool which can be used for the software implementation of certain digital signal processing algorithms. The advantages of DFT are: 1. It is computationally convenient. The DFT of a finite length sequence makes the frequency domain analysis much simpler than continuous Fourier transform technique. For an aperiodic signal x[n] the spectrum is: A X w xne jwn 1.

Because of uniform sampling, we have Each horizontal strip of the S-plane is overlaid onto the Z-plane to form the digital filter function from analog filter function. The frequency response of the digital filter is related to the frequency response of the Figure 6. Because of the aliasing that occurs in the sampling process, the frequency response of the resulting digital filter will not be identical to the original analog frequency response. To get the filter design procedure, let us consider the system function of the analog filter expressed in terms of a partial-fraction expansion It is important to recognize that the impulse invariant design procedure does not correspond to a mapping of the S-plane to the Z-plane.

A second approach to design of a digital filter is to approximate the derivatives in Eq. If the samples are closer together, the approximation to the derivative would be increasingly accurate. For example, suppose that the first derivative is approximated by the first backward difference Approximation to higher-order derivatives are obtained by repeated application of Eq.

We note that the operation 1 [ ] is a linear shift- invariant operator and that k [ ] can be viewed as a cascade of k operators 1 [ ]. It is easily verified that the left half of the S-plane maps into the inside of the small circle and the right half of the S-plane maps onto the outside of the small circle. Therefore, although the requirement of mapping the j-axis to the unit circle is not satisfied, this mapping does satisfy the stability condition.

Mapping of s-plane to z-plane corresponding to first backward-difference approximation to the derivative In contrast to the impulse invariance technique, decreasing the sampling period T, theoretically produces a better filter since the spectrum tends to be concentrated in a very small region of the unit circle. These two procedures are highly unsatisfactory for anything but low pass filters. An alternative approximation to the derivative is a forward difference and it provides a mapping into the unstable digital filters.

In the previous section a digital filter was derived by approximating derivatives by differences. An alternative procedure is based on integrating the differential equation and then using a numerical approximation to the integral. Consider the first - order equation Taking the Z-transform and solving for H z gives Solving Eq. In addition to the fact that the imaginary axis in the s-plane maps into the unit circle in the z-plane, the left half of the s-plane maps to the inside of the unit circle and the right half of the s-plane maps to the outside of the unit circle, as shown in Fig.

Thus we see that the use of the bilinear transformation yields stable digital filter from analog filter. Also this transformation avoids the problem of aliasing encountered with the use of impulse invariance, because it maps the entire imaginary axis in the s-plane onto the unit circle in the z-plane. The price paid for this, however, is the introduction of a distortion in the frequency axis. Mapping of the s-plane into the z-plane using the bilinear transformation 6. Another method for converting an analog filter into an equivalent digital filter is to map the poles and zeros of Ha s directly into poles and zeros in the z-plane.

For analog filter Thus each factor of the form s-a in Ha s is mapped into the factor 1- e aT z The eq 2 can be viewed as a computational procedure an algorithm for determining the output sequence y n of the system from the input sequence x n. Different realizations are possible with different arrangements of eq 2 The major issues considered while designing a digital filters are: Realiability causal or non causal Stability filter output will not saturate Sharp Cutoff Characteristics Order of the filter need to be minimum this leads to less delay Generalized procedure having single procedure for all kinds of filters Linear phase characteristics 1 1 1 0.

It must be a simple design b. There must be modularity in the implementation so that any order filter can be obtained with lower order modules.

Designs must be as general as possible. Having different design procedures for different types of filters high pass, low pass, is cumbersome and complex.

Cost of implementation must be as low as possible e. The important features of this class of filters can be listed as: Typical design procedure is analog design then conversion from analog to digital 7. The main features of FIR filter are, They are inherently stable Filters with linear phase characteristics can be designed Simple implementation both recursive and nonrecursive structures possible Free of limit cycle oscillations when implemented on a finite-word length digital system 7.

The group delay is defined as which is negative differential of phase function. Nonlinear phase results in different frequencies experiencing different delay and arriving at different time at the receiver. This creates problems with speech processing and data communication applications.

Having linear phase ensures constant group delay for all frequencies. The further discussions are focused on FIR filter. The section to follow will discuss on design of FIR filter.

Since linear phase can be achieved with FIR filter we will discuss the conditions required to achieve this. The roots of this polynomial constitute zeros of the filter. If the system impulse response has symmetry property i. The following points give an insight to this issue. This condition is suited in Band Pass filter design. Looking at these points, antisymmetric properties are not generally preferred. If zero is complex and z 1 then and we have two pairs of complex zeros.

As it can be seen there is pattern in distribution of these zeros. The standard methods of designing FIR filter can be listed as: Fourier series based method 2. Window based method 3. Frequency sampling method 7. Since the desired freq response Hd e j is a periodic function in with period 2, it can be expressed as Fourier series expansion This expansion results in impulse response coefficients which are infinite in duration and non causal.

It can be made finite duration by truncating the infinite length. The linear phase can be obtained by introducing symmetric property in the filter impulse response, i. It can be made causal by introducing sufficient delay depends on filter length 7.

From the desired freq response using inverse FT relation obtain hd n 2. Truncate the infinite length of the impulse response to finite length with assuming M odd 3.

Write the expression for H z ; this is non-causal realization 5. Design an ideal bandpass filter with a frequency response: Find H z. Design an ideal band reject filter with a frequency response: The arbitrary truncation of impulse response obtained through inverse Fourier relation can lead to distortions in the final frequency response.

The arbitrary truncation is equivalent to multiplying infinite length function with finite length rectangular window, i. The FT of w n is given by The whole process of multiplying h n by a window function and its effect in freq domain are shown in below set of figures.

The ideal sharp cutoff chars are lost and presence of ringing effect is seen at the band edges which is referred to Gibbs Phenomena.

This is due to main lobe width and side lobes of the window function freq response. The main lobe width introduces transition band and side lobes results in rippling characters in pass band and stop band. Smaller the main lobe width smaller will be the transition band. The ripples will be of low amplitude if the peak of the first side lobe is far below the main lobe peak. Increase length of the window - as M increases the main lob width becomes narrower, hence the transition band width is decreased -With increase in length the side lobe width is decreased but height of each side lobe increases in such a manner that the area under each sidelobe remains invariant to changes in M.

Thus ripples and ringing effect in pass-band and stop-band are not changed. Choose windows which tapers off slowly rather than ending abruptly - Slow tapering reduces ringing and ripples but generally increases transition width since main lobe width of these kind of windows are larger. Window having very small main lobe width with most of the energy contained with it i. Window design is a mathematical problem, more complex the window lesser are the distortions.

Rectangular window is one of the simplest window in terms of computational complexity. The different window functions are discussed in the following sention.

It is defined mathematically by, 7. This window function is given by, 7. The mathematical description is given by, 1 0 1 4 cos The mathematical description is given by, Type of window Appr. Among the window functions discussed Kaiser is the most complex 1 0 2 1 2 1 2 1 0 2 2 0 s s.

Obtain hd n from the desired freq response using inverse FT relation 2. Truncate the infinite length of the impulse response to finite length with assuming M odd choosing proper window 3. Design an ideal highpass filter with a frequency response: Design a filter with a frequency response: We know that DFT of a finite duration DT sequence is obtained by sampling FT of the sequence then DFT samples can be used in reconstructing original time domain samples if frequency domain sampling was done correctly.

The samples of FT of h n i. Since the designed filter has to be realizable then h n has to be real, hence even symmetry properties for mag response H k and odd symmetry properties for phase response can be applied.

Also, symmetry for h n is applied to obtain linear phase chas. Fro DFT relationship we have 1 , Since the impulse response samples or coefficients of the filter has to be real for filter to be realizable with simple arithmetic operations, properties of DFT of real sequence can be used.

The following properties of DFT for real sequences are useful: Exercise problems Prob 1: The filter should have linear phase and length of The typical frequency response characteristics is as shown in the below figure. As seen from differentiator frequency chars. Hilbert transformers are used to obtain phase shift of 90 degree. They are also called j operators. They are typically required in quadrature signal processing. Problem 3: If the realization can have less of these then it will be less complex computationally.

Any realization requiring less of these is preferred. No computing system has infinite precision. With finite precision there is bound to be errors.

The extent of this effect varies with type of arithmetic used fixed or floating. The serious issue is that the effects have influence on system characteristics. A structure which is less sensitive to this effect need to be chosen. The parallel processing can be in software or hardware. Longer pipelining make the system more efficient. FIR system is described by,. Direct form 2. Cascade form 3. Frequency-sampling realization 4.

Lattice realization 8. Realize the following using system function using minimum number of multiplication. The zeros of H z are grouped in pairs to produce the second order FIR system. In case of linear phase FIR filter, the symmetry in h n implies that the zeros of H z also exhibit a form of symmetry. Thus simplified fourth order sections are formed. Realize the difference equation 4 3 The term 1-z -N is realized as FIR and the term.

The realization of the above freq sampling form shows necessity of complex arithmetic. Upgrading filter orders is simple. Only additional stages need to be added instead of redesigning the whole filter and recalculating the filter coefficients.

These filters are computationally very efficient than other filter structures in a filter bank applications eg. Wavelet Transform 3. Lattice filters are less sensitive to finite word length effects. This is an indication that there isa zero on the unit circle. Direct form-I 2. Direct form-II 3. Cascade form 4. Parallel form 5. Lattice form 8. Typical Direct form I realization is shown below.

The upper branch is forward path and lower branch is feedback path. The number of delays depends on presence of most previous input and output samples in the difference equation.

We identify,.

M k k k z b z V z Y 1 1 all zeros The corresponding difference equations are,. The second order sections are required to realize section which has complex-conjugate poles with real co-efficients. Pairing the two complex- conjugate poles with a pair of complex-conjugate zeros or real-valued zeros to form a subsystem of the type shown above is done arbitrarily. There is no specific rule used in the combination. Although all cascade realizations are equivalent for infinite precision arithmetic, the various realizations may differ significantly when implemented with finite precision arithmetic.

If there are complex set of poles which are conjugative in nature then a second order section is a must to have real coefficients. Consider an All-pole system with system function. A general IIR filter containing both poles and zeros can be realized using an all pole lattice as the basic building block. Now the output is given by.

Consider a FIR filter with system function: Sketch the direct form and lattice realizations of the filter. Wavelet Transform 6. Flag for inappropriate content. Related titles. Ece v Digital Signal Processing [10ec52] Solution. Digital Signal Processing by J. Katre Tech Max. Digital signal processing by sk mitra 4th edition. Jump to Page. Search inside document. The form shown in eq 7 requires complex multiplications which can be avoided doing suitable modifications divide and multiply by 1 1 z W k N.

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