1. Digital Signal Processing
Prepared by:
Bhawna Bhardwaj
Assistant professor
B.P.R.C.E Gohana

2. Digital Signal Processing
Course at a glance
Discrete-Time
Signals &Systems
Fourier Domain
Representation
Sampling &
Reconstruction
System
Structure
Analysis
Z-Transform
DFT
Filter
Filter Structure
Filter Design

3. Digital Signal Processing
Chapter 1
Signals, Systems and Signal Processing.
Classification of Signals.
Concept of Frequency in Continuous-
Time & Discrete-Time Signals.
Analog to Digital & Digital to Analog Conversion.
Fourier transform

4. Digital Signal Processing
1.1. Signals, Systems and Signal Processing
Signal is defined as any physical quantity that varies with independent variables. For Example, the functions
S1(t) = 5t or S2(t) = 20t2 one variable
S(x,y) = 3x+4xy+6x two variables x and y
Speech signal
Frequency
Phase
Amplitude

5. Digital Signal Processing
1.1. Signals, Systems and Signal Processing
System, is defined as a physical device that performs an operation on a signal.
Basic elements of a digital signal processing system:
A/D Converter
Digital Signal
Processing
D/A Converter
Analog input
signal
Analog output
Digital input
Digital output

6. Digital Signal Processing
1.1. Signals, Systems and Signal Processing
Advantages of DSP
Flexibility (software change)
Accuracy
Reliable Storage
Complex process realized by simple code
Cost, Cheaper than analog

7. Digital Signal Processing
Classification of Signals
Continuous-Time versus Discrete-Time Signals:
Continuous-Time or analog signal are defined for every value of time.
are examples of analog signals
x(t) t
Analog Signal
Continuous in time.
Amplitude may take on any value in the continuous range of (-∞, ∞).
Analog Processing
Differentiation, Integration, Filtering, Amplification.
Implemented via passive or active electronic circuitry.

8. Digital Signal Processing
1.2. Classification of Signals
Continuous-Time versus Discrete-Time Signals:
Discrete-Time signals are defined only at certain specific value of time.
Continuous Amplitude.
Only defined for certain time instances.
Can be obtained from analog signals via sampling.
x(n)
Defined
The function provide an example of a discrete-time signal. -1 1 2 3 4 5 6 7 n
Undefined

9. Digital Signal Processing
1.2. Classification of Signals
Continuous-Valued versus Discrete-Valued Signals:
The values of a CT or DT Signal can be continuous or discrete.
If a signal takes on all possible values of a finite or an infinite range, it is CONTINUOUS-VALUED Signal.
If the signal takes on values from a finite set of possible values, it is DISCRETE-VALUED Signal. Also called Digital Signal because of the discrete values.
x(n) n 1 2 3 4 5 6 7 -1 8
Digital Signal with 4 different amplitude values

10. Digital Signal Processing
1.2. Classification of Signals
Deterministic versus Random Signals:
Deterministic Signal
Any signal whose past, present and future values are precisely known without any uncertainty
Random Signal
A signal in which cannot be approximated by a formula to a reasonable degree of accuracy (i.e. noise).

11. Fourier Transform
A CT signal x(t) and its frequency domain, Fourier transform signal, X(jw), are related by
For example:
Often you have tables for common Fourier transforms
The Fourier transform, X(jw), represents the frequency content of x(t).
analysis
synthesis

12. Fourier Transform of a Time Shifted Signal
We’ll show that a Fourier transform of a signal which has a simple time shift is:
i.e. the original Fourier transform but shifted in phase by –wt0
Proof
Consider the Fourier transform synthesis equation:
but this is the synthesis equation for the Fourier transform
e-jw0tX(jw)

13. Convolution in the Frequency Domain
We can easily solve ODEs in the frequency domain:
Therefore, to apply convolution in the frequency domain, we just have to multiply the two Fourier Transforms.
To solve for the differential/convolution equation using Fourier transforms:
Calculate Fourier transforms of x(t) and h(t): X(jw) by H(jw)
Multiply H(jw) by X(jw) to obtain Y(jw)
Calculate the inverse Fourier transform of Y(jw)

14. Proof of Convolution Property
Taking Fourier transforms gives:
Interchanging the order of integration, we have
By the time shift property, the bracketed term is e-jwtH(jw), so

15. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Conceptually, the A/D comprise 3 step process as in the following figure.
Sampler
Quantizer
Coder
Analog
signal
Digital
Discrete-Time
Quantized
x(n)
xa(t)
xq (n)
…..
A/D Converter

16. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Sampling:
It is the conversion of a CT signal into DT signal obtained by taking “Samples” of the CT signal at DT instants.
Periodic or Uniform Sampling:
This type of sampling is used most often in practice, describe by the relation:
where x(n) is the DT signal obtained by taking samples of the analog signal xa(t) every T seconds.
The rate at which the signal is sampled is Fs: Fs = 1/T
Fs is called the SAMPLING RATE or SAMPLING FREQUENCY (Hz)

17. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Sampling:
Consider an analog sinusoidal signal of the form:
Sampling Frequency:
Normalized frequency:
Sampled Signal:

18. Digital Signal Processing1
Introduction
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Sampling:
Relation among frequency variable:

19. Digital Signal Processing
1.4. A/D & D/A Conversion
4.1. Analog to Digital Converter (A/D):
Sampling:
We observe that the fundamental difference between CT and DT signals in their range of values of the frequency variables F and f or Ω and ω.
Means Sampling from infinite frequency range for F (or Ω) into a finite frequency range for f (or ω).
Since the highest frequency in a DT signal is ω = π or f = 1/2.
With sampling rate Fs the corresponding highest values of F and Ω are:

20. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Sampling:
Examples:
Two analog sinusoidal signals:
Which are sampled at a rate Fs = 40 Hz.
Discrete-time signals:
However,
This mean
The frequency F2 = 50 Hz is an alias of the frequency F1 = 10 Hz at the sampling rate of 40 samples per second.
F2 is not the only alias of F1

21. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Two analog sinusoidal signals, F1 = 1 Hz & F2 = 5 Hz are sampled at a rate Fs = 4 Hz.
F2 is the alias of F1

22. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Aliasing
Aliasing occurs when input frequencies (again greater than half the sampling rate) are folded and superimposed onto other existing frequencies.
In order to prevent alias
where Fmax is the highest input frequency
Nyquist Rate:
Minimum sampling rate to prevent alias.

23. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Sampling:
Sampling Theorem:
Given Band Limited (Frequency Limited Signal) with highest frequency Fmax: The signal can be exactly reconstructed provided the following is satisfied:
Sampling Frequency:
The samples are not quantized (analog amplitudes)

24. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Reconstruction Formula:
The signal:
The samples:
Formula:
Interpolation Function:

25. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Quantization:
The process of converting a DT continuous amplitude signal into digital signal by expressing each sample value as a finite number of digits is called QUANTIZATION.

26. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Quantization:
Fs = 1 Hz

27. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Quantization:
Numerical illustration of quantization with one significant digit using truncation or rounding

28. Digital Signal Processing
1.4. A/D & D/A Conversion
Analog to Digital Converter (A/D):
Coding:

29. Digital Signal Processing1
Introduction
1.4. A/D & D/A Conversion
Digital to Analog Converter (A/D):

30. Thank you