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

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

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

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+6x2 two variables x and y Speech signal Frequency Phase Amplitude

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

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

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.

Digital Signal Processing 1.2. Classification of Signals 1.2.2. 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

Digital Signal Processing 1.2. Classification of Signals 1.2.3. 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

Digital Signal Processing 1.2. Classification of Signals 1.2.4. 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).

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

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)

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)

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

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. 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) 0101101….. A/D Converter

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.1. 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)

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.1. Sampling: Consider an analog sinusoidal signal of the form: Sampling Frequency: Normalized frequency: Sampled Signal:

Digital Signal Processing 1 Introduction 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.1. Sampling: Relation among frequency variable:

Digital Signal Processing 1.4. A/D & D/A Conversion 4.1. Analog to Digital Converter (A/D): 1.4.1.1. 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:

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.1. 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

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. 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

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. 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.

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.1. 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)

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): Reconstruction Formula: The signal: The samples: Formula: Interpolation Function:

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.2. 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.

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.2. Quantization: Fs = 1 Hz

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.2. Quantization: Numerical illustration of quantization with one significant digit using truncation or rounding

Digital Signal Processing 1.4. A/D & D/A Conversion 1.4.1. Analog to Digital Converter (A/D): 1.4.1.3. Coding:

Digital Signal Processing 1 Introduction 1.4. A/D & D/A Conversion 1.4.2. Digital to Analog Converter (A/D):

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