DEPARTMENT OF ECE
TWO MARKS
DIGITAL SIGNAL PROCESSING
UNIT I
1. Define DFT of a discrete time sequence.
The DFT is used to convert a finite discrete time sequence x (n) to an N point
frequency domain sequence denoted by X (K). The N point DFT of a finite duration
sequence x (n) is defined as
N-1
X (K) = ∑ x (n) e-j2‫ה‬nk/N for K=0, 1, 2,…N-1
n=0
2. Define IDFT.
The IDFT is used to convert the N point frequency domain sequence X (K) to an
N point time sequence. The IDFT of the sequence X (K) of length N is defined as
N-1
x (n) =1/N ∑X (K) e+j2‫ה‬nk/N for n=0, 1,2,……N-1
K=0
3. List any four properties of DFT.
Let DFT{x (n)} =X (K), DFT{x1 (n)} =X1 (K), DFT{x2 (n)} =X 2(K)
i)
ii)
iii)
iv)
Periodicity: X (K+N) =X (K) for all K.
Linearity: DFT[a1 x1 (n)+a2 x2(n)]=a1 X1 (K)+a2 X2 (K)
DFT of time reversed sequence: DFT[ x(N-n)]=X(N-K)
Circular convolution :DFT[x1(n)*x2(n)]=X1(K) X2(K)
4. What is the relation between Z transform and DFT?
Let N point DFT of x (n) be X (K) and z transform of x (n) be X (Z)
The N point sequence X(K) can be obtained from X(Z) by evaluating X(Z) at N equally
spaced points around the unit circle .i.e X(K)=X(Z)/Z=ej 2 k‫ ה‬/N for K=0,1,2….(N-1)
5. State the shifting property of DFT
If DFT {x (n)} =X (K),
then DFT{x (n-no)} =X (K) e -j2‫ה‬no k/N
6. What is FFT?
The FFT (Fast Fourier Transform) is a method for computing the DFT with
reduced number of calculations. The computational efficiency is achieved by employing
divide and conquers approach. This is based on the decomposition of an N point DFT
into successively smaller DFT.
In an N point sequence if N can be expressed as N= r m then the sequence can be
decimated into r- point sequences. For each r- point sequence, r point DFT is computed.
From the results of r- point DFTs the r 2 - point DFTs are computed. From the results of
r 2- point DFT the r3- point DFT are computed and so on, until we get r m point DFT.
Hence the number of stage of computation is m. The number r is called the radix of the
FFT algorithm.
7. What is radix- 2 FFT?
The radix- 2 FFT is an efficient algorithm for computing N point DFT of an N
point sequence. In radix- 2 FFT the N point sequence is decimated into 2 -point sequence
and the 2 point DFT for each decimated sequence is computed. From the result of 2point DFT the 4- point DFT are computed. From the result of 4- point DFT the 8- point
DFT are computed. From the result of 4- point DFT, the 8 point DFT are computed and
so on until we get N point DFT.
8. How many multiplication and addition are involved in radix- 2 FFT?
The total number of complex addition is N log N and the total numbers of
complex multiplication are N/2log N.
9. How many multiplication and addition are involved in DFT?
The total numbers of complex additions are N (N-1) and the total number of
complex multiplications are N 2
10. What is twiddle factor (or) phase factor?
The complex number WN is called phase factor or twiddle factor.
WN=e-j2‫ח‬/N
11. Draw and explain the basic butterfly diagram (or) flow graph of DIT radix-2
FFT?
The basic butterfly diagram of DIT radix-2 FFT is shown below. It performs the
following operations .Here a & b are the input complex number and A&B are output
complex number.
i) Input complex number b is multiplied by the phase factor WN K.
ii) The product bWNK is added to the input complex number a to from a new
complex number A.
iii) The product bWNK is subtracted to the input complex number a to from a new
complex number B.
12. What is DIT radix-2 FFT?
The DIT (Decimal in Time) radix-2 FFT is an efficient algorithm for computing
DFT. In DIT radix-2 FFT, the decimal N-point sequence is decimated into 2-point
sequences. The results of 2-point DFT are used to compute 4-point DFTs. Two numbers
of 2-point DFT are combined to get a 4-point DFT. The results of 4-point DFTs are used
to compute 8-point DFTs. Two numbers of 4-point DFTs are combined to get an 8-point
DFT. This process is continued until we get N-point DFT.
13. What is DIF radix-2 FFT?
The DIF radix -2 FFT is an efficient algorithm for computing DFT. In this
algorithm the N point time domain sequence is converted to two numbers of N/2 point
sequences. Then each N/2 point sequence is converted to two number of N/4 point
sequences. This process continued until we get N/2 numbers of 2 point sequences. Now
the 2 point DFT of N/2 number of 2 point sequence will give N samples, which is the N
point DFT of time domain sequence .Here the equation for forming N/2 point
sequences,N/4 sequences etc., are obtained by decimation of frequency domain
sequences. Hence this method is called DIF.
14. Draw and explain the basic butterfly diagram or flow graph of dif radix2 FFT?
The basic butterfly diagram of DIF radix2 FFT is shown below. It performs the
following operations .Here a & b are the input complex number and A&B are output
complex number.
i) The sum of the complex number a and b are computed to form a new complex
number A.
ii) The complex number b is subtracted from a to get the difference a-b.
iii) The difference term a-b is multiplied with phase factor WNK to from a new complex
number B.
15. Compare the DIT and DIF.
DIT radix-2 FFT
1.When the input is bit reversed order, the
output will be in normal order .
2.In each stage of computation the phase
factor are multiplied before add and
subtract operation
3.The value of N should be expressed such
that N=2 m and this algorithm consists of
m stage of computation.
4.Total number of arthemetric operations is
N log N complex addition and N/2logN
complex multiplications.
DIF radix-2 FFT
1.When the input is normal order, the
output will be in bit reversed order .
2.In each stage of computation the phase
factor are multiplied after add and subtract
operation
3.The value of N should be expressed such
that N=2 m and this algorithm consists of
m stage of computation.
4.Total number of arithmetic operations is
N log N complex addition and N/2logN
complex multiplications
16. What are the applications of FFT algorithm?
i) Linear filtering.
ii) Correlation.
iii)Spectrum analysis.
17. Calculate the number of multiplications needed in calculation of DFT and FFT
with 64 point Sequence.
DFT
The number of complex multiplications required using direct computation is
N2=642=4096
FFT
The number of complex multiplication required using FFT is
N/2logN=64/2log64=192
18. Calculate the number of addition needed in calculation of DFT and FFT with
16 point Sequence.
DFT
The number of complex addition required using direct computation is
N(N-1)=16(16-1)=240
FFT
The number of complex addition required using FFT is
N log N=16log16=64
19. What is main advantage of FFT?
FFT reduces the computation time required to compute Discrete Fourier
Transform.
20. How we can calculate IDFT using FFT algorithm?
The inverse DFT of an N point sequence X (K); K=0, 1…N-1 is defined as
N-1
x (n) =1/N ∑ X (K) e+j2‫ה‬nk/N for n=0, 1,2,…N-1
K=0
Take complex conjugate and multiply by N, we get
N-1
Nx *(n) = ∑X *(K) e+j2‫ה‬nk/N for n=0, 1, 2 …N-1
K=0
The desired output sequence x (n) can then be obtained by complex conjugating
the DFT and divided by N
N-1
x (n) =1/N [ ∑X* (K) e+j2‫ה‬nk/N ]*
K=0
21. Find the DFT of the following signal:x(n)=δ(n)
N-1
X (K) = ∑ x (n) e-j2‫ה‬nk/N for K=0, 1, 2….,N-1
n=0
N-1
X (K) = ∑ δ (n)e-j2‫ה‬nk/N for K=0, 1, 2,…N-1
n=0
=1
22. What are the properties of convolution?
i) Commutative law: x(n)*h(n)=h(n)*x(n)
ii) Associative law:[x(n)*h1(n)]*h2(n)=x(n)*[h1(n)*h2(n)]
iii)Distributive law :x(n)*[h1(n)+h2(n)] =x(n)*h1(n)+x(n)*h2(n)
23. The first five DFT coefficients of a sequence x(n)
are X(0)=20,X(1)=5+2j,X(2)=0,X(3)=0.2+0.4j,x(4)=0.Determining the remaining
DFT coefficients.
X(5)=X*(3)=0.2-0.4j
X(6)=X*(2)=0
X(7)=X*(1)=5-2j
UNIT II
1. What are the types of digital filter according to their impulse response?
i)IIR (Infinite Impulse Response) filter
ii)FIR ( Finite Impulse Response) filter
2. What are the advantages of FIR filter?
1. FIR filter have exact linear phase.
2. FIR filters are always stable.
3. FIR filter can be realized in both recursive and recursive structures.
4. Excellent design methods are available for various kinds of FIR filter.
5. FIR filter are free of limit cycle oscillation, when implemented on a finite word
length digital system.
3. What are the disadvantages of FIR filter?
i) Memory requirement and execution time are very high.
ii The implementation of narrow transition band fir filters is very costly, as it
requires considerably more arithmetic operation and hardware components
such as multipliers, adders and delay elements.
4. What is the necessary and sufficient condition for the linear phase characteristic
of a FIR filter?
The phase function should be a linear function of w, in which requires constant
group delay and phase delay.
θ(w)=-ζw
for satisfying above condition
h (n)=h(N-1-n)
i.e. The impulse response must be symmetrical about ζ =(N-1)/2
If only constant group delay is desired then
θ(w)=β-ζw
for satisfying above condition
h (n)=-h(N-1-n)
i.e. The impulse response must be symmetrical about ζ =(N-1)/2
5. Distinguish IIR and FIR filters
FIR
IIR
Impulse response is finite
Impulse Response is infinite
They have perfect linear phase
Non recursive
They do not have perfect linear
phase
Recursive
Greater flexibility to control the
shape of magnitude response
Less flexibility, usually limited
to specific kinds of filters.
6. List the well known design technique for linear phase FIR filter design?
1. Fourier series method
2. Window method
3. Frequency sampling method.
7. What is Gibbs phenomenon?
OR
What are Gibbs oscillations?
One possible way of finding an FIR filter that approximates H d(e j)would be
to truncate the infinite Fourier series at n=  (N-1/2).Abrupt truncation of the
series will lead to oscillation both in pass band and is stop band .This
phenomenon is known as Gibbs phenomenon.
8. What are the desirable characteristics of the windows?
The desirable characteristics of the window are
1. The central lobe of the frequency response of the window should contain
most of the energy and should be narrow.
2. The highest side lobe level of the frequency response should be small.
3. The side lobes of the frequency response should decrease in energy
rapidly as w tends to П
9. Under what conditions a finite duration sequence h(n) will yield constant group
delay in its frequency response characteristics and not the phase delay?
If the impulse response is anti symmetrical, satisfying the condition
h (n)=-h(N-1-n)
The frequency response of FIR filter will have constant group delay and not the
phase delay .
10. Give the equation specifying rectangular window function.
The weighing function for the rectangular window is given by
WR (n) = {1
0
-(N-1)/2≤n≤(N-1)/2
otherwise
11. Give the equation specifying hamming window function.
The weighing function for the rectangular window is given by
WH (n) = { 0.54+0.46cos (2 П n /N-1)
0
- (N-1)/2≤n≤ (N-1)/2
otherwise
12. Compare Hamming window with Rectangular Window.
Hamming window
Rectangular Window
1.The main lobe width is equal to 8/N
and the peak side lobe level is –41dB.
1.The main lobe width is equal to 4/N
and the peak side lobe level is –13dB.
2.The low pass FIR filter designed will
have minimum stop band attenuation
of –53 dB
2.The low pass FIR filter designed will
have minimum stop band attenuation
of –21dB
13. Compare Hamming window with Kaiser Window.
Hamming window
Kaiser window
1.The main lobe width is equal to 8/N
and the peak side lobe level is –41dB.
The main lobe width, the peak side lobe
level can be varied by varying the
parameter  and N.
2.The low pass FIR filter designed will
have first side lobe peak of –53 dB
The side lobe peak can be varied by
varying the parameter .
14. Write the steps involved in FIR filter design.
i) Choose the desired frequency response H d(w).
ii) Take the inverse Fourier transform and obtain hd(n)
iii) Convert the infinite duration sequence hd(n) to h(n)
h(n)=hd(n)*w(n)
iv) Take Z transform of h (n) to get H (Z).
v) Substitute z=e-jw, find the frequency response H (e-jw)
15. What is the reason that FIR filter is always stable?
FIR filter is always stable because all its poles are at the origin.
16. What condition on the FIR sequence h (n) are to be imposed N order that this
filter can be called a liner phase filter?
The conditions are
(i)
Symmetric condition h(n)=h(N-1-n)
(ii)
Anti symmetric condition h(n)=-h(N-1-n)
17. What are the advantages of Kaiser Window?
1. It provides flexibility for the designer to select the side lobe level and N.
2. It has the attractive property that the side lobe level can be varied continuously
from the low value in the Blackman window to the high value in the rectangle
Window.
18. What is the principle of designing FIR filter using frequency sampling method?
In frequency sampling method, a set of sample is determined from the desired
frequency response and are identified as discrete Fourier transform coefficients. The
inverse discrete Fourier transform of this set of samples then gives the filter
coefficients. The set of sample points used in this procedure can be determined by
sampling a desired frequency response Hd (ejw) at N points wk, k=0,1,….N-1
uniformly spaced around the unit circle.
19. How to design an FIR filter?
To design a filter means to select the coefficients such that the system has specific
characteristics. The required characteristics are stated in filter specifications. Most of the
time filter specifications refer to the frequency response of the filter. There are different
methods to find the coefficients from frequency specifications:
Window design method
Frequency Sampling method
20. How to characterize digital FIR filters
There are a few terms used to describe the behavior and performance of FIR filter
including the following:




Filter Coefficients - The set of constants, also called tap weights, used to multiply
against delayed sample values. For an FIR filter, the filter coefficients are, by
definition, the impulse response of the filter.
Impulse Response – A filter’s time domain output sequence when the input is an
impulse. An impulse is a single unity-valued sample followed and preceded by
zero-valued samples. For an FIR filter the impulse response of a FIR filter is the
set of filter coefficients.
Tap – The number of FIR taps, typically N, tells us a couple things about the
filter. Most importantly it tells us the amount of memory needed, the number of
calculations required, and the amount of "filtering" that it can do. Basically, the
more taps in a filter results in better stopband attenuation (less of the part we want
filtered out), less rippling (less variations in the passband), and steeper rolloff (a
shorter transition between the passband and the stopband).
Multiply-Accumulate (MAC) – In the context of FIR Filters, a "MAC" is the
operation of multiplying a coefficient by the corresponding delayed data sample
and accumulating the result. There is usually one MAC per tap.
21. What are "FIR filters"?
FIR filters are one of two primary types of digital filters used in Digital Signal Processing
(DSP) applications (the other type being IIR).
22. What does "FIR" mean?
"FIR" means "Finite Impulse Response".
23. Why is the impulse response "finite"?
The impulse response is "finite" because there is no feedback in the filter; if you put in an
impulse (that is, a single "1" sample followed by many "0" samples), zeroes will
eventually come out after the "1" sample has made its way in the delay line past all the
coefficients.
24. How do I pronounce "FIR"?
Some people say the letters F-I-R; other people pronounce as if it were a type of tree. We
prefer the tree. (The difference is whether you talk about an F-I-R filter or a FIR filter.)
25. What is the alternative to FIR filters?
DSP filters can also be "Infinite Impulse Response" (IIR). (See dspGuru's IIR FAQ.) IIR
filters use feedback, so when you input an impulse the output theoretically rings
indefinitely.
26. How do FIR filters compare to IIR filters?
Each has advantages and disadvantages. Overall, though, the advantages of FIR filters
outweigh the disadvantages, so they are used much more than IIRs.
UNIT III
1.. How can you design a digital filter from analog filter?
Digital filter can de designed from analog filter using the following methods
1. Impulse invariant method
2. Bilinear transformation
2. State the steps to design digital IIR filter using bilinear method.
Substitute s by 2/T (z-1/z+1), where T=2/Ώ (tan (w/2) in H(s) to get H (z).
3. Give the bilinear transform equation between s plane and z plane
s=2/T (1-z-1/1-z+1)
4. What is bilinear transformation?
Bilinear transformation is a one to one mapping from the s-domain to the zdomain. That is, the bilinear transformation is a conformal mapping that transforms the
j Ω axis into the unit circle in the z plane only once, thus avoiding the aliasing of
frequency components. Also the transformation of a stable analog filter result in a stable
digital filter as all the poles in the left half of the s plane are mapped inside the unit circle
of the z plane.
5. What is warping effect?
The relation between the analog and digital frequencies in bilinear transformation
is given by
Ω=2/T tan w/2
For smaller values of w there exist linear relationship between w and Ω.but for
larger values of w the relationship is nonlinear. This introduces distortion in the
frequency axis. This effect compresses the magnitude and phase response. This effect is
called warping effect
6. Write a note on pre warping.
The effect of the non linear compression at high frequencies can be compensated.
When the desired magnitude response is piecewise constant over frequency, this
compression can be compensated by introducing a suitable rescaling or
prewarping the critical frequencies by using the formula
Ω=2/T tan w/2
7. Why impulse invariant method is not preferred in the design of IIR filters other
than low pass filter?
In this method the mapping from s plane to z plane is many to one. i.e. ,all the
poles in the s plane between the intervals (2k-1)∏/T to (2k+1)∏/T .Thus there are an
infinite number of poles that map to the same location in the z plane, producing an
aliasing effect. Due to spectrum aliasing the impulse invariant method is inappropriate in
designing high pass filters. That is why the impulse method is not much preferred. in the
design of IIR filters other than low pass filter
8. By impulse invariant method obtain the digital filter transfer function and the
differential equation of the analog filter H(s) =1/s+1
S- pK=1-epKTz-1
S+1=0
S= -1= pK
Substitude the value of pK
H (z) =1/1-e-Tz-1
9. What is meant by impulse invariant method?
In this method of digitizing an analog filter, the impulse response of the resulting
digital filter is a sampled version of the impulse response of the analog filter. For e.g. if
the transfer function is of the form,
N
H(S)=∑ 1/s-pk, then
K=1
N
H (z) =∑1/1-epKTz-1
K=1
10. Give the magnitude function of Butterworth filter. What is the effect of varying
order N on magnitude and phase response?
The magnitude function of Butterworth filter is
|H(jΏ)=1/[1+(Ώ/Ώc)2N]1/2 ,N=1,2,3,4,….
Where N is the order of the filter and Ώc is the cutoff frequency. The magnitude
response of Butterworth filter closely approximates the ideal response as the order N
increases. The phase response becomes more non linear as N increases.
11. Give any properties of butterworth filters.
i)The magnitude response of butterworth filter closely approximates the ideal
response as the order N increases
ii) The magnitude response of butterworth filter decreases monotonically as the
frequency Ώ increases from 0 to ∞
iii)the Poles of the butterworth filter lies on a unit circle.
12. Write the expression for order of Butterworth filter?
The expression is N=log (λ /€) /log (1/k)
Where
k= Ώp/ Ώs
λ= (100.1αs-1)0.5
€= (100.1αp-1)0.5
13. What are IIR filters? What does "IIR" mean?
IIR filters are one of two primary types of digital filters used in Digital Signal Processing
(DSP) applications (the other type being FIR). "IIR" means "Infinite Impulse Response".
14. Why is the impulse response "infinite"?
The impulse response is "infinite" because there is feedback in the filter; if you put in an
impulse (a single "1" sample followed by many "0" samples), an infinite number of nonzero values will come out (theoretically).
15. What is the alternative to IIR filters?
DSP filters can also be "Finite Impulse Response" (FIR). FIR filters do not use feedback,
so for a FIR filter with N coefficients, the output always becomes zero after putting in N
samples of an impulse response.
16. What are the advantages of IIR filters (compared to FIR filters)?
IIR filters can achieve a given filtering characteristic using less memory and calculations
than a similar FIR filter.
17. What are the disadvantages of IIR filters (compared to FIR filters)?



They are more susceptable to problems of finite-length arithmetic, such as
noise generated by calculations, and limit cycles. (This is a direct
consequence of feedback: when the output isn't computed perfectly and is
fed back, the imperfection can compound.)
They are harder (slower) to implement using fixed-point arithmetic.
They don't offer the computational advantages of FIR filters for multirate
(decimation and interpolation) applications.
18. Butterworth filter –
no gain ripple in pass band and stop band, slow cutoff
19. Chebyshev filter(Type I) –
no gain ripple in stop band, moderate cutoff
20. Chebyshev filter(Type II) –
no gain ripple in pass band, moderate cutoff
21. Bessel filter
- no group delay ripple, no gain ripple in both bands, slow gain cutoff
22. Elliptic filter –
gain ripple in pass and stop band, fast cutoff
23. How do I skip needless calculations?
First, if your filter has zero-valued coefficients, you don't actually have to calculate those
taps; you can leave them out. A common case of this is "half-band" filter, which have the
property that every-other coefficient is zero.
Second, if your filter is "symmetric" (linear phase), you can "pre-add" the samples which
will be multiplied by the same coefficient value, prior to doing the multiply. Since this
technique essentially trades an add for a multiply, it isn't really useful in DSP
microprocessors which can do a multiply in a single instruction cycle. However, it is
useful in ASIC implementations (in which addition is usually much less expensive than
multiplication); also, some newer DSP processors now offer special hardware and
instructions to make use of this trick.
24. What is Impulse response ?
The impulse response is a mathematical concept that can be approximated in the real
world. It is the output of a circuit when an ideal impulse (zero width pulse with unit area)
is applied to the input. The Laplace transform deals with this.
The spectrum of an ideal impulse is flat and so the frequency shape of the resulting
output of the network is the frequency response of the network
UNIT IV
1. What are the different types of arithmetic in digital systems?
There are three types of arithmetic used in digital system
1. Fixed point arithmetic
2. Floating point arithmetic
3. Block floating arithmetic.
2. What do you understand by a fixed-point number?
In fixed point arithmetic the position of the binary point is fixed. The bit to the
right represents the fractional part of the number & those to the left represent the integer
part. For example, the binary number 01.1100 has the value 1.75 in decimal.
3. What are the different types in fixed-point number representation?
Depending on the way negative number is represented, there are three different
forms of fixed-point arithmetic. They are
1. Sign-magnitude
2.1's complement
3. 2's complement
4. What do you understand by 2's complement representation?
In two's complement representation positive number are represented as in sign
magnitude & 1's complement. The negative number is obtained by complementing all the
bits of the positive number & adding one to the least significant bit.
For example
(0.5625)10 = (0.100100)2
(-0.5625)10 = 1.011011
0.000001
------------(1.011100)2
5. What is meant by block floating point representation? What are its advantages?
In block point arithmetic the set of signals to be handled is divided into blocks.
Each block has the same value for the exponent. The arithmetic operations with in the
block uses fixed point arithmetic & only one exponent per block is stored thus saving
memory. This representation of numbers is more suitable in certain FFT flow graph & in
digital audio applications.
6. What are the advantages of floating point arithmetic?
1. Large dynamic range
2. Over flow in floating point representation is unlikely.
7. What are the three-quantization errors to finite word length registers in digital
filters?
1. Input quantization error
2. Coefficient quantization error
3. Product quantization error
8. How the multiplication & addition are carried out in floating point arithmetic?
In floating point arithmetic, multiplication are carried out as follows,
Let f1 = M1*2c1 and f2 = M2*2c2.
Then f3 = f1*f2 = (M1*M2) 2(c1+c2)
That is, mantissa is multiplied using fixed-point arithmetic and the exponents are
added.
The sum of two floating-point numbers is carried out by shifting the bits of the
mantissa of the smaller number to the right until the exponents of the two numbers
are equal and then adding the mantissas.
9. What do you understand by input quantization error?
In digital signal processing, the continuous time input signals are converted into
digital using a b-bit ACD. The representation of continuous signal amplitude by a fixed
digit produce an error, which is known as input quantization error.
10. What is product quantization error?
Product quantization error arises at the output of a multiplier. Multiplication of b
bit data with a b bit coefficient result a product having 2b bits. Since b bit register is used
the multiplier output must be rounded or truncated to b bits, which produce an error. This
error is known as quantization error.
11. What is coefficient quantization error?
The filter coefficient is computed to infinite precision theory. But in digital
computation the filter coefficient are represented in binary and are stored in registers.
If b bit register is used the filter coefficient must be rounded or truncated to b bit which
produce an error. Due to quantization of coefficient the frequency response of filter may
differ appreciably from the desired response and some time the filter may actually fail to
meet the desired specifications. If the poles of desired filter are close to the unit circle,
then those of the filter with quantized coefficients may lie just outside the unit circle
leading to unstability.
12. What are the different quantization methods?
The common methods of quantization are
1. Truncation 2. Rounding
13. What is the relationship between truncation error e(n) and the bits b for
representing a decimal into binary?
For a 2's complement representation, the error due to truncation for both positive and
negative values of x is
-2-b≤ e (n) ≤0
Where b is the number of bits
14. What is meant rounding? Discuss its effect on all types of number
representation?
Rounding a number to b bits is accomplished by choosing the rounded result as the b
bit number closest to the original number unrounded.
For fixed point arithmetic, the error made by rounding a number to b bits satisfy the
inequality
-2-b
2-b
----- ≤ e (n) ≤ -------2
2
for all three types of number systems, i.e., 2's complement, 1's complement & sign
magnitude.
For floating point number the error made by rounding a number to b bits satisfy the
inequality
-2-b≤e (n) ≤ 2-b
where e (n) =xT-x
15. What is truncation?
Truncation is a process of discarding all bits less significant than least significant
than least significant bit that is retained.
0.00110011 to 4 bit 0.0011.
16. What is meant by A/D conversion noise?
A DSP contains a device, A/D converter that operates on the analog input x (n) to
produce xq(n) which is binary sequence of 0s and 1s.
At first the signal x(t) is sampled at regular intervals to produce a sequence x(n) is of
infinite precision. Each sample x(n) is expressed in terms of a finite number of bits given
the sequence xq(n). The difference signal e (n)=xq(n)-x(n) is called A/D conversion
noise.
17. What is the effect of quantization on pole location?
Quantization of coefficients in digital filters lead to slight changes in their value.
This change in value of filter coefficients modify the pole-zero locations. Some times the
pole locations will be changed in such a way that the system may drive into instability.
18. Which realization is less sensitive to the process of quantization?
Cascade form.
19. What is meant by quantization step size?
Let us assume a sinusoidal signal varying between +1 and -1 having a dynamic
range 2. If the ADC used to convert the sinusoidal signal employs b+1 bits including sign
bit, the number of levels available for quantizing x(n) is 2 b+1. Thus the interval between
successive levels
q = 2 / 2 b+1 =2-b
Where q is known as quantization step size.
20. How would you relate the steady-state noise power due to quantization and the b
bits representing the binary sequence?
Steady state noise power
σe 2 =2-2b/12
Where b is the number of bits excluding sign bit.
21. What are the two kinds of limit cycle behavior in DSP?
1. Zero input limit cycle oscillations
2. Overflow limit cycle oscillations
22. What is overflow oscillation?
The addition of two fixed-point arithmetic numbers cause over flow the sum exceeds
the word size available to store the sum. This overflow caused by adder make the filter
output to oscillate between maximum amplitude limits. Such limit cycles have been
referred to as over flow oscillations.
23. What is meant by limit cycle oscillation?
For an IIR filter, implemented with infinite precision arithmetic the output should
approach zero in the steady state if the input is zero .however the non linearity due to the
finite precision arithmetic operation often cause periodic oscillations to occur in the
output. Such oscillation in recursive systems are called zero input limit cycle oscillations.
24. What are the methods used to prevent overflow?
There are two methods used to prevent overflow
1. Saturation arithmetic 2. Scaling
25. Define "dead band" of the filter
The limit cycle occur as a result of quantization effect in multiplication. The
amplitudes of the output during a limit cycle are confined to a range of values called the
dead band of the filter.
26. Explain briefly the need for scaling in the digital filter implementation.
To prevent overflow, the signal level at certain points in the digital filter must be
scaled so that no overflow occurs in the adder.
27. Why the limit cycle problem does not exist when FIR digital filter is realized in
direct form or cascade form?
In case of FIR filter there are no limit cycle oscillations, if the filter is realized in
direct form or cascade form since these structures have no feedback.
28. Why rounding is preferred to truncation in realizing digital filter?
1. The quantization error due to rounding is independent of type of arithmetic.
2. The mean of rounding error is zero.
3. The variance of the rounding error signal is low.
29. What is the steady state variance of the noise in the output due to quantization of
the input for the first order filter?
y(n)=ay(n-1)+x(n)
σe 2 =(2-2b/12)(1/1-a2)
30. Compare the fixed point and floating point arithmetic.
Fixed point arithmetic.
Floating point arithmetic.
Fast operation
Slow operation
Relatively economical
More expensive because of costlier
hardware.
Increased dynamic range
Round off error occur with both addition
and multiplication.
Overflow occur in addition
Small dynamic range
Round off error occur only for addition
Overflow occur in addition
31. Express the fraction 7/8 and -7/8 in sign magnitude, 2’s complement and 1’s
complement.
Fraction (7/8) = (0.111)2 in sign magnitude, 2’s complement and 1’s complement.
Fraction (-7/8)= (1.111)2 in sign magnitude
(1.000)2 1’s complement.
(1.001)2 2’s complement.
32. Give the expression for signal to quantization noise ratio and calculate the
improvement with an increase of 2 bit to the existing bit.
SNR=6.02b+10.79+10log10 σx 2
With an increase of 2 bits, increase in SNR is approximately 12 Db
UNIT V
1. What is the energy density spectrum?
The quantity |X a(F)|2 represent the distribution of signal energy as a function of
frequency and hence it is called the energy density spectrum of the signal that is
Sxx(F)=|X(F)|2
∞
Sxx(f)= |∑ x(n)e-j2ПKf |2
n= -∞
2. What is power density spectrum?
Let x(t) be a stationary random process. The statistical autocorrelation function
for this signal is
γxx(τ)=E[x*(t)x(t+τ)]
The Fourier transform of the autocorrelation function of a stationary random process
gives their power density spectrum
γxx(f)=F(γxx(τ))
∞
=∫ γxx(τ) e-j2Пf τ dτ
-∞
3. Using indirect method, how to find the energy density spectrum?
It requires two steps:
i)
First, the autocorrelation rxx(k) is computed from x(n)
ii)
The Fourier transform of the autocorrelation is computed.
∞
r xx(k)= ∑x*(n)x(n+k)
n= -∞
∞
Sxx(f)= ∑ r xx(k)e-j2ПKf
n= -∞
4. What is three types of non parametric methods of power spectrum estimation?
i). Bartlett methods (Averaging periodogram)
ii) Welch method (modified Averaging periodogram)
iii) Blackman and tukey methods (smoothing periodogram)
5. What are nonparametric methods?
These methods make no assumption about how the data were generated and hence
are called nonparametric methods.
6. Define Periodogram.
Schuster defined the periodogram as a method to discover the frequencies of the
hidden harmonics signal. The estimate Pxx (f) can also be expressed as
N-1
Pxx(f)=1/N | ∑ x(n) e-j2Пfn |2 =1/N|X(f)|2
n= 0
Where X(f) is the Fourier transform of the sample sequence x(n).This well
known form of the power density spectrum estimate is called the Periodogram.
7. What is a Bartlett method?
In this method to reduce the variance of the Periodogram,
Three steps
i) First divide the N point sequence x (n) into k non overlap subsequence of length M
ii) Find the periodogram for each sub sequence.
iii) Calculate the average periodogram of k subsequence.
8. What is the advantage of Bartlett window methods?
The effect of reducing the length of data from N point to M=N/K, result in a
window whose spectral width has been increased by a factor of K. consequently, the
frequency resolution has been reduced by a factor K. in return in resolution we reduced
the variance.
9. What is the difference between Bartlett and welch’s methods?
i) First difference is that welch’s method allows overlapping of data sequence. The
overlap 50% or 75%.
ii) The second difference is the data with in a sequence are windowed prior to computing
the periodogram.
10. What is a Blackman and turkey method?
Blackman and turkey proposed a method in which the sampled autocorrelation
sequence is windowed first and then Fourier transformed.
11. Define quality of nonparametric methods?
The ratio of it’s the square of the mean of power spectrum estimate to variance.
Qa=E[Pxx(f)]2/var[Pxx(f)]
12. Define variability.
The reciprocal of this quantity called the variability.
13. What is the quality of three power spectrum estimate?
Estimate
Bartlett
Welch
Blackman-tukey
Quality factor
1.11N∆f
1.39N∆f
2.34 N∆f
14. Determine the frequency resolution of Bartlett, Welch, Blackman turkey
methods of
power spectrum estimate. For quality factor 10.assume that overlaps in Welch method
is 50% and length of the sample sequence is 1000.
Quality factor=10
Length of sample sequence (N) =1000
Overlap in Welch method=50%
Bartlett method
QB=1.11N∆f
∆f= (Qbart/1.11N)=0.009
Welch method
Qw=1.39N∆f
∆f=0.0072
Blackman-turkey
QBT=2.34 N ∆f
∆f=0.0042
15. What is the advantage of Welch methods?
i) The Welch periodogram is reasonably computationally efficient due to the use
of FFT algorithms.
ii) The Welch method the variance of the random process is reduced compared to
basic periodogram and Bartlett methods.
iii) Welch method allows all the windowing techniques.
16. Limitation of non parametric methods for power spectrum estimation? (What
are the disadvantages of non-parametric methods of power spectral estimation?)
i) It requires long data sequence to obtain the necessary frequency resolution.
ii) Spectral leakage effect because of windowing.
iii)The assumption of the autocorrelation estimator x(m) to be zero for m≥N. This
assumption limits the frequency resolution and quality of power spectrum estimate.
iv)Assumption that the data are periodic with period N. These assumptions may not be
realistic.
17. Define mean.
The procedure of determining the average weight of a group of objects by
summing their individual weights and dividing by the total number of objects gives the
average value of x..
Mathematically the discrete sample mean can be described
_
n
X =1/n ∑xi
i=1
For the continuous case that mean value of the random variable
X is defined as
∞
X =E[X] =∫xfX(x) dx
-∞
where E[X] is read ``the expected value of X''. Other names for the same mean
value x or the expected value E[X] are average value and statistical average.
18. What are the two properties of power spectrum estimator?
i) Bias
ii) Variance
19. Define the bias of estimator?
Bias of estimator is defined as the true value of the parameter minus the expected
value of the estimator.
20. Define unbiased estimator.
An unbiased estimator is one for which the bias is 0. This then means that the
expected value of the estimator is the true value so that the probability density is
symmetrical then its center would be at the true value.
21. What is consistent estimator?
An estimator is said to be consistent if as the number of observation becomes
lager, the bias and the variance both tends to zero.
If the bias and variance both tend to zero as the limit tends to infinity or the
number of observations become large, the estimator is said to be consistent.
22. What is Variance?
The variance of an estimator effectively measures the width of the probability
density and is defined as
σmx=E[(mx)2]-[ E[(mx)]2
In using estimates the mean value estimate of mx, for a Gaussian random process
is the sample mean.A good estimator should have a small variance in addition to having a
small bias suggesting that the probability density function is concentrated about its mean
value. This says that as the number of observations N increase, the variance of the sample
mean decreases, and since the bias is zero, the sample mean is a consistent estimator.
23 .What is deterministic and random signals with an example.
Deterministic signals are functions that are completely specified in time. The
nature and amplitude of such signal at any time can be predicted. The Patten of the signal
is a regular and can be characteristics mathematically.
Examples
i)
x (t)=ct this is the ramp whose amplitude of this signal increases linearly
with time and slope is c.
Random Signal (Non Deterministic Signal)
A non deterministic signal is one whose occurrence is random in nature and its
Patten is quite irregular.
Example:
i) A typical example of a non deterministic signal is thermal noise in an electrical
circuit.
24. Define the terms: i) auto correlation ii) cross correlation
Correlation gives a measure of similarity between two data sequences.
Auto correlation
For sequence x(n) the auto correction function r xx(k) is defined as
∞
r xx(k)= ∑x*(n)x(n+k)
k=0,±1,±2,.....
n= -∞
or equivalently,
∞
r xx(k)= ∑x*(n)x(n-k)
k=0,±1,±2,.....
n= -∞
Cross correlation
For two sequence x(n) and y(n),the cross correction function r xy(k) is defined as
∞
r xy(k)= ∑x*(n)y(n+k)
n= -∞
or equivalently,
∞
r xy(k)= ∑x*(n)y(n-k)
n= -∞
k=0,±1,±2,.....
k=0,±1,±2,.....
25. Define auto covariance.
Related to the autocorrelation function is the auto covariance function, which is
defined as
Cxx (t1, t2) =E [xt1-m (t1)] [xt2-m (t2)]
=γxx (t1, t2)-m(t1)m(t2)
Where m (t1) =E [xt1] and m (t2) =E [xt2] are the mean of x1 and x2
respectively. When the process is stationary
Cxx(t1,t2)=Cxx(t1-t2)=Cxx(τ)= γxx (τ)-mx2
Where τ = t1-t2
26. What is zero padding? Does zero padding improve the frequency resolution in
Spectral estimate?
Zero padding is increase the length of sequence by adding zero to the given
sequence.
Note that zero padding does not change the resolution but it does have the effect
of interpolating the spectrum pxx(f)
27. Define cross power spectral density.
The definition of the power density spectrum can be extended to two jointly
stationary random process x(t) and y(t),which have a cross correction function γxy(τ)
The fourier transform of the γxy(τ) is
∞
γxy(F) =∫ γxy(τ) e-j2ПF τ d τ
-∞
Which is called the cross power density spectrum.