Digital Signal Processing EEE343/ECE328: Lecture-1 Digital Signal Processing

DSP research areas Audio signal processing, audio compression, Digital image processing, video compression, Speech processing, speech recognition, Digital communications, RADAR, SONAR,  Neural Signal Processing DNA sequencing For solving communication technology related issues And many more…

Advantages of Digital over Analog Signal Processing Flexibility in reconfiguring Accuracy Storage facility in magnetic media (tape or disc) Cost minimization Transportability of digital signals

Signals, Systems and Signal Processing Any physical quantity that varies with time, space or any other independent variable/s. Mathematically, a signal is a function of one or more variable/s: s1(t)= 5t s2(t)= 20t2 s3(x,y)= 3x+2xy+ 10y2 Above are the examples of precisely defined signals. Examples of natural signals: Speech, Electrocardiogram (ECG)

Signals, Systems and Signal Processing A physical device that performs an operation on a signal. Usually, signal generation is also associated with a system that responds to a force. Example: Filter (reduces noise and interference) Signal Processing: Systems are characterized by the operation it performs on the signal that are passed through it. Such operations are usually referred to as signal processing.

Classification of Signals Continuous-Time Versus Discrete-Time Signals Continuous –Time Signal: Discrete–Time Signal:

Sampling of Analog Signals: Where, T = Sampling period 1/T = FS = Sampling rate/freq Relationship between time variables t and n of CT and DT signals: t = nT = n/Fs

Classification of Signals (Contd.) Continuous-Valued Versus Discrete-Valued Signals Continuous Valued - Values can be taken from any finite\infinite range Discrete Valued - Values taken from finite range Discrete Time + Discrete Valued = Digital Signal

Classification of Signals (Contd.) Deterministic Versus Random Signals Deterministic - Uniquely described by an explicit mathematical expression, a table of data or well defined rule - All past , present and future values are known Random - Noise signal

The concept of frequency in discrete-time signals Closely related to periodic motion (harmonic oscillation) Has the dimension of inverse time The nature of time affect the nature of frequency accordingly Continuous-Time Sinusoidal Signals: Mathematically: xa(t)= Acos(Ωt+θ), -∞<t< ∞ xa(t)= an analog signal A= amplitude of the sinusoid Ω = frequency in radian/s θ= phase in radians F (cycles per second or hertz (Hz)) is also used where, Ω=2πF In terms of F, xa(t) can be expressed as: xa(t) = Acos(2πFt+θ), -∞<t< ∞

Properties of continuous-time sinusoidal signals For every fixed value of the frequency F, xa(t) is periodic xa(t+Tp)= xa(t) where Tp =1/F is the fundamental period of the signal Continuous-time sinusoidal signals with distinct frequencies are themselves distinct Increasing F results in an increase in the rate of oscillation of the signal more periods in given time interval

xa(t)= Acos(Ωt+θ)= ½Aej(Ωt+θ) + ½ Ae-j(Ωt+θ) Properties (Contd.) Sinusoidal signals can be represented by complex exponentials as: xa(t)= Aej(Ωt+θ) From Euler identity: e ±jφ = cosφ ± jsinφ So, the sinusoidal signal xa(t) can be expressed as: xa(t)= Acos(Ωt+θ)= ½Aej(Ωt+θ) + ½ Ae-j(Ωt+θ) Note: Concept of negative frequency is introduced for convenience of calculation

Discrete-Time Sinusoidal Signals Mathematical expression: x(n)= Acos(ωn+θ), -∞<n< ∞ n = integer variable, called the sample number A= amplitude of the sinusoid ω = frequency in radian/sample θ= phase in radians Instead of ω we use f defined by, ω=2πf Then x(n)= Acos(2πfn+θ), -∞<n< ∞ f has dimensions of cycles per sample.

Properties of Discrete-Time Sinusoidal Signals By definition, x(n) is periodic with period N (N> 0) if and only if x(n+N)= x(n) for all n The smallest value of N, for which the above relation is true is called the fundamental period For a sinusoid with frequency f0 to be periodic, we should have cos[2π f0 (N+n)+θ] = cos(2π f0 n+θ) =>cos(2π f0 N+2π f0n+θ) = cos(2π f0 n+θ) =>2π f0 N=2πk =>f0 N=k =>f0 =k/N This relation is true if and only if there exists an integer k. Note: Slight change in frequency can result in large change in the period, f1= 31/60, here time period is 60, but f2= 30/60=1/2, has a time period N=2

So, A discrete-time sinusoid is periodic, only if its frequency f is a rational number

xk(n) = Acos(ωkn+θ), k= 0,1,2…. Properties (Contd.) Discrete-time sinusoids whose frequencies are separated by an integer multiple of 2π are identical Considering the sinusoid, cos(ω0n+θ), we can say, As a result, all sinusoidal sequences, xk(n) = Acos(ωkn+θ), k= 0,1,2…. where ωk=ω0+ 2kπ, -π ≤ω 0 ≤ π are identical.

Sinusoidal sequences with frequencies in the range -π ≤ ω0 ≤ π are distinct. Discrete-time sinusoidal signals with frequencies |ω|< π or |f|< ½ are unique. This range is called the fundamental range

Properties (Contd.) The highest rate of oscillation in a discrete-time sinusoid is attained when ω= π or equivalently, f= ½ Let us consider the sinusoidal signal, x(n) = cos ω0n, Where, we take value of ω0 = 0, π/8, π/4, π/2, π corresponding to f= 0, 1/16, 1/8, ¼, ½, for which periods are N= ∞, 16, 8,4, 2.

Properties (Contd.) The period of the sinusoid decreases as the frequency increases, 0 to π whereas, the rate of oscillation increases. To see what happens for π ≤ ω0 ≤ 2π, we consider the sinusoids with frequencies ω1 =ω0 and ω2 = 2π - ω0. As ω1 varies from 0 to π, ω2 varies from π to 0. Thus x1(n) = A cosω1n = A cosω0n x2(n) = A cosω2n = A cos (2π - ω0)n = A cos (- ω0)n = x1(n) Hence, x2(n) is an alias x1(n). The result would be same for sine function, except for phase difference between x1(n) and x2(n). With the increasing frequency from π to 2π, the rate of oscillation decreases. For ω0 = 2π, the result is a constant signal, as in the case for ω0 = 0 Obviously, for ω0 = π (or f = ½ ) we have the highest rate of oscillation.

Analog to digital conversion Most useful signals are analog: Speech, biological signals, seismic signals, radar signals, sonar signal, audio and video signals etc. For processing, conversion of these signals into digital form is necessary. Conversion process is called analog-to-digital (A/D) conversion and the corresponding devices are called A/D converters (ADCs) Conceptually, the process has three steps: 1. Sampling 2. Quantization 3. Encoding A/D converter xa(t) x(n) xq(t) 01011… Sampler Quantizer Coder Discrete-Time Signal Quantized Signal Analog Signal Digital Signal

Analog to digital conversion 1. Sampling: Conversion of continuous–time signal to discrete-time signal by taking “samples” of continuous time signal at discrete instants. where, T is called the sampling interval 2. Quantization: Conversion of a discrete-time continuous-valued signal into a discrete-time discrete-valued (digital) signal. Value of each signal sample is represented by a value from a finite set of possible values. Quantization error= x(n)-xq(n) 3. Coding: Each discrete value xq(n) is represented by a b-bit binary sequence. Sampler xa(t) xa(nT) x(n)

Sampling of Analog Signals: Where, T = Sampling period 1/T = FS = Sampling rate/freq Relationship between time variables t and n of CT and DT signals: t = nT = n/Fs

Sampling of Analog Signals (Contd): To establish relationship between f/ω and F/Ω, Suppose, = A cos (2πfn + θ) So, Or, For CT: For DT:

Sampling of Analog Signals (Contd): Substituting the value of f with F/Fs, we find that freq. range of CT Sinusoid when sampled at a rate Fs = 1/T must fall in the range: With a sampling rate Fs, the Corresponding highest values of F and Ω are

Sampling of Analog Signals (Contd): Why the range of F for corresponding Fs is needed??? Let us assume two analog sinusoidal signals, For Fs = 40Hz, the corresponding DT signals are,

Sampling of Analog Signals (Contd):

Sampling of Analog Signals (Contd):

The sampling rate Fs = 2Fmax is called the Nyquist rate The Sampling Theorem: To avoid the problem of aliasing, Fs is selected so that Fs>Fmax Nyquist theorem states , If the highest frequency contained in an analog signal xa(t) is Fmax and the signal is sampled at a rate Fs>2Fmax then xa(t) can be exactly recovered from its sample values using the interpolation The sampling rate Fs = 2Fmax is called the Nyquist rate

The Sampling Theorem (Contd): Consider the following analog signals xa1(t) = 3 cos50πt+ 10 sin300πt – cos100πt xa2(t) = 13 cos25πt+ 8 cos75πt + 4 cos35πt xa3(t) = sin150πt+ 5 cos470πt – 3cos600πt What are the Nyquist rates for these signals?

The Sampling Theorem (Contd): Example 1.4.4 Consider the analog signal xa(t)= 3 cos 2000πt+ 5 sin 6000πt+ 10 cos 12000πt What is the Nyquist rate for this signal? Assume that, we sample this signal using a sampling rate Fs = 5000 samples/s. What is the discrete-time signal obtained after sampling? What is the analog signal ya(t) we can reconstruct from the samples if we use ideal interpolation?

The Sampling Theorem (Contd): Solution: xa(t)= 3 cos 2000πt+ 5 sin 6000πt+ 10 cos 12000πt a) The frequencies present in the analog signal are: F1= 1kHz, F2= 3kHz, F3= 6kHz Thus, Fmax= 6kHz, so according the Smapling Theorem, FN= 12kHz b)Since we have chosen, Fs= 5kHz, the folding frequency is, Fs/2=2.5kHz x(n) = xa(nT) = xa (n/Fs) = 3 cos2π(1/5)n+ 5 sin2π(3/5)n + 10 cos2π(6/5)n = 3 cos2π(1/5)n+ 5 sin2π(1-2/5)n + 10 cos2π(1+1/5)n = 3 cos2π(1/5)n+ 5 sin2π(-2/5)n + 10 cos2π(1/5)n x(n) = 13 cos2π(1/5)n - 5 sin2π(2/5)n

The Sampling Theorem (Contd): Problems: 1.5, 1.6,1.7, 1.8, 1.9, 1.10,1.11

Quantization of Continuous-Amplitude Signals: The process of converting a discrete-time continuous-amplitude signal into a digital signal by expressing each sample value as a finite number of digits (instead of infinite), is called quantization. Q[x(n)] denotes the quantization operation on the samples of x(n) and the quantized sample sequence at the output of the quantizer is denoted by xq(n). Hence, xq(n) = Q[x(n)] Quantization Error: eq(n) =xq(n) – x(n)

Quantization of Continuous-Amplitude Signals (Contd): Let us consider a DT signal, T= 1 sec

Quantization of Continuous-Amplitude Signals (Contd): TABLE: Quantization using Truncation or Rounding n x(n) Discrete-time Signal xq(n) (Rounding) eq(n) = xq(n) - x(n) (Rounding) (allowing 5% error) 1 1.0 0.0 0.9 2 0.81 0.8 -0.01 3 0.729 0.7 -0.029 4 0.6561 0.0439 5 0.59049 0.6 0.00951 6 0.531441 0.5 -0.031441 7 0.4782969 0.0217031 8 0.43046721 0.4 -0.03046721 9 0.387420489 0.012579511

Quantization of Continuous-Amplitude Signals (Contd): Quantization step or Resolution: Where, xmax = max value of x(n) (=1 in exmpl) xmin = min value of x(n) (=0 in exmpl) xmax – xmin = dynamic range L = number of quantization levels (=11 in exmpl) The quantization error cannot exceed half of the quantization step or resolution:

Quantization of Sinusoidal Signals (Contd):

Coding of Quantized Samples: If we have L levels we need at least L different binary numbers. With a word length of b bits we can create 2b different binary numbers. 2b ≥ L b ≥ log2L