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Exercise Questions

Q. y(n)-0.4 y(n-1) = x(n). find the anti-causal impulse response? h(n)=0 for n≥ 0

h(n-1) = 2.5 [h(n)- δ (n) ]

h(-1) =  2.5 [h(0)- δ (0) ] = -2.5

h(-2) = -2.52  . …….. h(n) = -2.5valid for n ≤ -1

Q. x(n)={1,2,3} y(n)={3,4} Obtain difference equation from i/p & o/p information 

y(n) + 2 y(n-1) + 3 y(n-2) = 3 x(n) + 4 x(n-1) (Ans)

Q. x(n) = {4,4,}, y(n)= x(n)- 0.5x(n-1). Find the difference equation of the inverse system. Sketch the realization of each system and find the output of each system.

Solution: 

The original system is y(n)=x(n)-0.5 x(n-1)

The inverse system is x(n)= y(n)-0.5 y(n-1)

y (n) = x (n) – 0.5 x(n-1)

Y (z) = X (z) [1-0.5Z-1]

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

System

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Inverse System

y (n) – 0.5 y(n-1) =x(n)

Y (z) [1-0.5 Z-1] = X (z)

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

g (n) = 4 δ (n) - 2δ(n-1) + 4δ (n-1) - 2δ (n-2) = 4δ (n) + 2δ (n-1) - 2δ (n-2)

y (n) = 0.5 y(n-1) + 4δ (n) + 2δ (n-1) – 2δ (n-2)

y (0) = 0.5y(-1) + 4δ (0) = 4

y(1) = 4

y(2) = 0.5 y(1) - 2δ (0) = 0

y(n) = {4, 4} same as i/p

Non Recursive filtersRecursive filters

y(n) Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

for causal system

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

For causal i/p sequence 

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Present response is a function of the present and past N values of the excitation as well as the past N values of response. It gives IIR o/p but not 

y(n) =  Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE) ak x(n-k)

Present response depends only on present i/p & previous i/ps but not future i/ps. It gives FIR o/p

always.

(n) – y(n-1) = x(n) – x(n-3)

Q. y(n) = 1/3[x (n+1) + x (n) + x (n-1)]    Find the given system is stable or not?

Let x(n) = δ (n)

h(n) = 1/3 [ δ (n+1) + δ (n) + δ(n-1)]

h(0) = 1/3

h(-1) = 1/3

h(1) = 1/3

s=∑h(n) <∞ therefore Stable.

Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE)

Q. y(n) = a y(n-1) + x(n)   given y(-1) = 0

Let x(n) = δ (n)

h(n) = y(n) = a y(n-1) + δ (n)

h(0) = a y(-1) +δ (0) = 1 = y(0)

h(1) = a y(0) + δ (1) = a

h(2) = a y(1) + δ (2) = a2 . . . . . . . h(n) = an u(n)   stable if a<1.

y(n-1) =1/a [ y(n) – x(n)]

y(n) = 1/a [ y(n+1) – x(n+1)]

y(-1) = 1/a [ y(0) – x(0)]=0

y(-2) = 0

 

Q. y(n) = 1/n+1 y(n-1) + x(n) for n ≥ 0

= 0        otherwise. Find whether given system is time variant or not?

Let x(n) = δ (n)

h (0) = 1 y(-1) +δ (0) = 1

h(1) = ½ y(0) + δ(1) = ½

h(2) = 1/6

h(3) = 1/24

if x(n) = δ (n-1)

y(n) = h(n-1)

h(n-1) = y(n) =1/n+1 h(n-2) + δ (n-1)

n=0 h(-1) = y(0) = 1 x 0+0 =0

n=1 h(0) = y(1) = ½ x 0 +δ (0)= 1

n=2 h(1) = y(2) = 1/3 x 1 + 0 = 1/3

h(2) = 1/12

∴ h (n, 0) ≠ h (n,1)  ∴ TV

Q. y (n) = 2n x(n)    Time varying

Q. y (n) =1/3 [x (n+1) + x (n) + x (n-1)] Linear

Q. y (n) = 12 x (n-1) + 11 x(n-2) TIV

Q. y (n) = 7 x2(n-1) non linear

Q. y (n) = x2(n) non linear

Q. y (n) =  n2 x (n+2) linear

Q. y (n) = x (n2) linear

Q. y (n) =  ex(n) non linear

Q. y (n) =  2x(n) x (n) non linear, TIV

(If the roots of characteristics equation are a magnitude less than unity. It is a necessary & sufficient condition) Non recursive system, or FIR filter are always stable

Q. y (n) + 2 y2(n) = 2 x(n) – x(n-1)  non linear, TIV

Q. y (n) - 2 y (n-1) = 2x(n) x (n) non linear, TIV

Q. y (n) + 4 y (n) y (2n) = x (n) non linear, TIV

Q. y (n+1) – y (n) = x (n+1) is causal

Q. y (n) - 2 y (n-2) = x (n) causal

Q. y (n) - 2 y (n-2) = x (n+1) non causal

Q. y (n+1) – y (n) = x (n+2) non causal

Q. y (n-2) = 3 x (n-2) is static or Instantaneous.
Q. y (n) = 3 x (n-2) dynamic

Q. y (n+4) + y (n+3) = x (n+2) causal & dynamic

Q. y (n) = 2 x (αη )

If α=1 causal, static

α <1 causal, dynamic

α >1 non causal, dynamic

α≠ 1 TV

Q. y (n) = 2(n+1) x (n) is causal & static but TV.

Q. y (n) = x (-n) TV 

The document Exercise Questions - Introduction to Digital Signal Processing | Digital Signal Processing - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Digital Signal Processing.
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FAQs on Exercise Questions - Introduction to Digital Signal Processing - Digital Signal Processing - Electronics and Communication Engineering (ECE)

1. What is digital signal processing in electrical engineering?
Ans. Digital signal processing (DSP) is a branch of electrical engineering that focuses on the analysis, manipulation, and synthesis of digital signals. It involves the use of mathematical algorithms to convert analog signals into digital form and perform various operations on them, such as filtering, noise reduction, compression, and modulation/demodulation.
2. How is digital signal processing different from analog signal processing?
Ans. Digital signal processing differs from analog signal processing in several ways. While analog signal processing deals with continuous signals, DSP deals with discrete signals represented by sequences of numbers. DSP also allows for precise control over signal properties, such as frequency and amplitude, using mathematical operations. Additionally, DSP offers the advantage of easier storage, manipulation, and transmission of signals compared to analog processing.
3. What are some applications of digital signal processing in electrical engineering?
Ans. Digital signal processing finds numerous applications in electrical engineering. It is used in audio and video processing, speech recognition and synthesis, radar and sonar systems, image processing, telecommunications, medical imaging, control systems, and many more. DSP algorithms are implemented in various devices, including smartphones, digital cameras, music players, and medical equipment.
4. What are the key components of a digital signal processing system?
Ans. A digital signal processing system typically consists of three main components: an analog-to-digital converter (ADC) to convert analog signals into digital form, a digital signal processor (DSP) or microcontroller to perform the required mathematical operations on the digital signals, and a digital-to-analog converter (DAC) to convert the processed digital signals back into analog form.
5. What skills are essential for a career in digital signal processing in electrical engineering?
Ans. A career in digital signal processing requires proficiency in mathematics, especially in areas such as calculus, linear algebra, and probability theory. Knowledge of programming languages like MATLAB, Python, or C/C++ is also essential for implementing DSP algorithms. Additionally, a strong understanding of signal processing concepts and familiarity with DSP hardware/software tools are valuable skills for success in this field.
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