CPE658 Digital Signal Processing

1. Chapter 1: Introduction CPE658 Digital Signal Processing Bundit Thipakorn, Ph.D. Computer Engineering Department

2. Signals and Systems 1.1 To study and analyze some physical phenomena, we need to determine the mean allowing us to understand and describe such phenomena in a systematic way. In practice, we can accomplish this goal by describing a physical phenomenon as a mathematical function called “Signal”. We encounter so many signals in our daily life which are generated by natural means. For example, the speech or the sound coming into our ears is a mechanical signal representing the air pressure. The picture that we see is a light signal representing the intensity of light. All electrical devices are associated with voltage and current signals.

3. S(t) = at S(t) = at2 + bt + c (1.1) S(t) = Asin(wt+f) S(x,y) = ax + by + cxy Definition 1: Signalis a pattern of variations of a measurable quantity that is a function of one or more independent variables such as time (t) and space (x and y). Signal can be represented mathematically as: Where a, b, c, A, w, and f are constant values. The function which is used to describe a signal is called the representation of the signal.

4. Typically, a signal carries information about the behavior or nature of the phenomenon. There are 3 components to signal theory: • Modeling: A process to determine a representation of the signal. • Analysis: A process to extract information carried by the signal (Signal Processing). • Design: A process to synthesize a physical process that is described by the signal. Examples of signals can be shown as following:

5. 4 x 10 1.5 1 0.5 0 -0.5 -1 0 200 400 600 800 1000 1200 1400 1600 1800 2000 A Speech Signal A Speech Waveform Amplitude Time Figure 1.1 Speech waveform. A speech signal is a mechanical signal representing the air pressure and carries voice information.

6. A Digital Image (2-D Signal) Figure 1.2 Digital image (2-D signal). The intensity I(x,y) at any location of a digital image is a function of two spatial independent variables (x and y). A digital image is a light signal representing the light intensity and carries visual information .

7. 1.1.1 Types of Signals 1. Continuous-Time ( c-t or analog ) Signal Amplitude Time Figure 1.3 Continuous-time (c-t) signal. A continuous-time (c-t) signal is a signal that is present for all instants in time or space.

8. 2.Discrete-Time (d-t) Signal Amplitude Time Figure 1.4 Discrete-time (d-t) signal. A discrete-time (d-t) signal is a signal that is present only at certain specific values of time or space.

9. 3.Digital Signal Amplitude Time Figure 1.5 A digital signal A digital signal is a discrete-time signal having a set of discrete values.

10. 1.1.2 Systems Figure 1.6 A block diagram representation of a continuous-time system. System is a physical device that performs an operation on a signal. We can consider a system as anything that takes an input signal, operates on it, and produces an output signal. x(t) y(t) System t [ ] Output (or Response) signal Input (or Excitation) signal Let x(t) and y(t) be the input and output signals, respectively, of a system. The system can be represented mathematically as: y(t) = t [x(t)] (1.2)

11. Figure 1. 7 A block diagram representation of an analog signal processing system. 1.1.3 Signal Processing Signal processing is a method to extract useful information carried by the signal. There are two types of signal processing systems: analog signal processing and digital signal processing. 1. Analog signal processing system. This is a system that processes the input signal directly on its analog form. Analog Signal Processor C-T output signal C-T input signal

12. 2. Digital signal processing system. The digital signal processing system, on the other hand, is a system that processes the input signal on its digital form. Fig. 1.8 shows a block diagram of the digital signal processing system. To perform the processing digitally, the digital signal processing system requires two additional steps. The first step is the step to convert a continuous-time signal into a discrete-time discrete valued (digital) signal which is called the analog-to-digital (A/D) conversion. A digital signal produced from the digital signal processor is converted back to continuous-time form by the other additional step called the digital-to-analog (D/A) conversion.

13. C-T input signal Analog-to- Digital Converter Digital Signal Processor Digital-to- Analog Converter Figure 1.8 A block diagram representation of a digital signal processing system. Digital signal Digital signal C-T output signal

14. Analog-to-Digital Conversion 1.2 Analog Signal Discrete-Time Signal Digital Signal Figure 1.9 A two-step process to generate a digital signal from an analog signal. The digital signal is generated from the analog (continuous-time) signal using these following two-step process: 1. Sampling Process 2. Quantization Process A sampling process is the process to sample a continuous-time (c-t) signal at a certain period of time called the sampling interval.

15. Thus, the sampling process will convert a c-t signal into a discrete-time (d-t) signal. (1.3) i.e. Where T = the sampling period Fs = the sampling frequency = 1/T n = 0, ±1, ±2,… x(n) = xa(nT) xa(t) Fs = 1/T Analog Signal Discrete-time Signal Sampler Figure 1.10 Periodic sampling of an analog signal.

16. A quantization process is the process to round up the values of the d-t signal to a finite set of possible values. Thus, the quantization process will convert a d-t continuous-valued signal into a d-t discrete-valued (digital) signal. 1.2.1 Digitization A digitization process is the process to convert a analog signal into an encoded digital signal. This process is usually called analog-to-digital (A/D) conversion and is illustrated in this Fig. 1.11.

17. Sampler Quantizer Encoder Figure 1.11 A block diagram of a digitization process. Analog signal Discrete-time continuous-valued signal Discrete-time discrete-valued (digital) signal Encoded-digital signal

18. The Concept of Frequency in Continuous-Time and Discrete-Time Signals 1.3 xa(t) Acos(q) t Tp Figure 1.12 An analog sinusoidal signal. An analog sinusoidal signal xa(t) can be represented as: xa(t) = Acos(Wt + q) (1.4) where

19. 1 F W = the angular frequency (rad/sec) = 2pF F = the frequency (cycles/sec) Tp = the fundamental period (sec) = q = the phase The analog sinusoidal signal described by Eq. 1.3 has these following properties: 1. Periodical: i.e. xa(t + Tp) = xa(t) (1.5) 2. Distinction: C-t sinusoidal signals with different frequencies are themselves distinct. 3. The rate of oscillation of the signal will be increased if the frequency F is increased.

20. x(n) = Acos(wn + q) n Figure 1.13 A discrete-time sinusoidal signal. A discrete-time sinusoidal signal x(n) can be represented as: x[n] = Acos(wn + q) (1.6) Where w = the angular frequency (rad/sample) = 2pf

21. f = the frequency (samples/sec) q = the phase The discrete-time sinusoidal signal described by Eq. 1.6 has these following properties: 1. A discrete-time sinusoid is periodic only if the frequency of the d-t signal is a rational number. By definition, a d-t signal x(n) is periodic with period N (N>0) if an only if x[n + N] = x[n] for all n (1.7) The smallest possible N satisfying the above condition is called “fundamental period” of the d-t signal.

22. Where w0 = the fundamental angular frequency f0 = the fundamental frequency If x[n] = Acos(w0n + q) , x[n+N] will be defined as: x[n+N] = Acos(w0(n+N) + q) (1.8) = A[cos(w0n+ q)cosw0N - sin(w0n+ q)sinw0N)] = 2pf0 Eq. 1.8 will be equal to Eq. 1.6 only if sinw0N = 0 and cosw0N = 1 which can be satisfied if and only if:

23. 2pk N w0 = k N f0 = w0N = 2pk Where k = any integer number i.e. For a d-t sinusoidal x[n] with frequency f0 to be periodic, this following condition must exist for any integer k: or (1.9)

24. 2. Non-distinction: Discrete-time sinusoids that have frequencies separated by an integer multiple of 2p are identical. Hence the frequency range for d-t sinusoids is finite with duration 2p. Usually, the range: is used and it is called the fundamental range. 3. At w=p or -p (f=0.5 or -0.5), the highest frequency in a discrete-time signal is obtained.

25. To establish the relationship between the frequency F (or W) of analog signals and the frequency f (or w) of d-t signals, we start from considering an analog sinusoidal signal expressed as shown in Eq. (1.4): xa(t) = Acos(Wt + q) When we sample this analog signal at a rate of Fs samples per second, the d-t signal x(n) can be expressed as following: xa(nT) x(n) = Acos(WnT + q) (1.10) By comparing Eq. (1.10) with Eq. (1.4), the relationship between the angular frequency of a d-t signal w and the angular frequency of an analog signal W can be expressed as shown in Eq. (1.11):

26. w = WT (1.11) or (1.12) f = F/Fs The conversion from the analog frequency to the digital frequency or the conversion from the digital frequency to the analog frequency can be summarized as illustrated in Table 1.1. From Table 1.1, we can notice that the basic difference between c-t and d-t signals is in their range of frequency values. C-T signals have the infinite frequency range. D-T signals, on the other hand, have the finite frequency range. Thus there is a possibility that more than one analog frequencies are mapped into the same digital frequency.

27. _ _ _ _ _ _ _ _ < < < < < < < < - - < W < < F < Table 1.1 Relations among frequency variables C-T Signals D-T Signals W = 2pF w = 2pf w = WT and f = F/Fs W = w/T and F = fFs -p w p 1/2 -1/2 f -p/T W p/T Fs/2 -Fs/2 F

28. wmax = p or fmax = 1/2, The highest frequency in c-t signal depends on the sampling frequency. Since the highest frequency in a d-t signal is: (1.13) the corresponding highest values of analog frequency must be: (1.14) or (1.15)

29. An analog signal xa(t) can be reconstructed from its sample values xa(nT) if the sampling rate 1/T is greater than twice the highest frequency Fmax presenting in xa(t). Definition: The sampling rate 2Fmax for an analog band-limited signal is referred to as the Nyquist rate. Therefore, any frequency above Fs/2 or below -Fs/2 results in samples that are identical to corresponding frequency in the range: The Sampling Theorem

30. From the sampling theorem, we must sample a c-t signal with Fs > 2Fmax to ensure that all the sinusoidal components in the c-t signal are mapped into corresponding d-t frequency components with frequency in the fundamental interval. Thus, we must have some knowledge about the frequency content of any given analog signal in order to sample it with an appropriate sampling rate. However, this detailed knowledge of the characteristics of such signals is normally not available prior to obtaining the signal. In fact, it is the information that we would like to extract in digital signal processing.

31. Classification of Signals 1.3 1.3.1 Energy and Power Signals Let x[n] be a d-t signal. The total energy, E, and power, P, of x[n]over an infinite time interval time interval can be determined as following: (1.16) and (1.17)

32. A signal x[n] is called an “energy signal” if and only if the total energy is finite which means the power is equal to zero. A signal x[n] is called a “power signal” if and only if the total energy is infinite and the power is finite. i.e. E P Energy signal 0 Power signal

33. 1.3.2 Periodic and Aperiodic Signals In case of a d-t signal, a signal x[n] is periodic with period N (N>0) if and only if: x[n+kN] = x[n] for all n (1.18) where k is any integer. The smallest value of N for which Eq. 1.18 holds is called the “fundamental period”. On the other hand, if there is no value of N satisfying Eq. 1.18, the signal is called an “aperiodic” (non-periodic) signal.

34. 1.3.3 Classification Based on Symmetry A signal x[n] is called a conjugate-symmetric signal if x[n] = x*[-n] (1.19) On the other hand, signal x[n] is called a conjugate- antisymmetric signal if x[n] = -x*[-n] (1.20) In the case of real-value signal, a conjugate- symmetric signal is called an “even signal” and a conjugate-antisymmetric signal is called an “odd signal”. Thus

35. Even signal: x[n] = x [-n] (1.21) Odd signal: x[n] = -x[-n] (1.22) From Eq. 1.22, x[0[ = -x[0]; therefore, an odd signal must be 0 at n=0. Any complex signal x[n] can be expressed as a sum of its conjugate- symmetric part xcs[n] and its conjugate-antisymmetric part xca[n] as show in Eq. 1.23. x[n] = xcs[n] + xca[n] (1.23)

36. 1 1 1 1 xcs[n] = (x[n] + x*[-n]) xod[n] = (x[n] - x[-n]) xca[n] = (x[n] - x*[-n]) xev[n] = (x[n] + x[-n]) 2 2 2 2 where (1.24) and (1.25) Similarly, any real signal x[n] can be expressed as a sum of its even part xev[n] and its odd part xod[n] as show in Eq. 1.26. x[n] = xev[n] + xod[n] (1.26) where (1.27) and (1.28)

37. Basic Operation on Signals 1.4 An example of the even signal is a sinusoidal signal expressed as the cosine function. A sinusoidal signal expressed as the sine function is an example of the odd signal. Typically, in the area of signal processing, it is required some basic operations to perform a simply process such as addition, multiplication, amplify, etc. on the signals. 1.4.1 A Signal Multiplier A signal multiplier will form the product of values of x1[n] and x2[n] signals at each instant as illustrated in Fig. 1.14.

38. Figure 1.14 A signal multiplier. For a signal multiplier having a sinusoidal signal as one of its input, this operation will be called “modulation”. The device performs the modulation operation is known as a “modulator”. 1.4.2 An Adder An adder will form the addition of values of x1[n] and x2[n] signals at each instant as illustrated in Fig. 1.15.

39. Figure 1.15 An adder. 1.4.3 A Scalar Multiplier An output y(t) of a scalar multiplier will be the result of multiplication a input signal x(t) with a scalar A as illustrated in Fig. 1.16. Figure 1.16 A scalar Multiplier

40. 1.4.4 A Time Shifting An operation that performs a time delaying and advancing on signal is known as “time-shifting” operation. Let x[n] be an input signal and y[n] be an output signal resulting from a time-shifting operation. y[n] will be defined as following: y[n] = x[n - N] (1.29) where t0 is the time shift and N is an integer N > 0 Delaying (Shift to the right) Advancing (Shift to the left) N < 0

41. 1.4.5 A Time Reversal A time reversal, which sometimes is called the folding or the reflection operation, is the operation to replace the independent variable “n” by “-n” as shown in Eq. 1.30. (1.30) y[n] = x[-n] Let TD and TR represent the time-shifting and time-reversal, respectively. Thus, we can represent TD and TR as shown in Eq. 1.31. TDk[x[n]] = x[n-k] k > 0 (1.31) TR[x[n]] = x[-n]

42. The combination operation of TD and TR can be expressed as either Eq. 1.32 or Eq. 1.33. We can notice that the result of performing TR before TD (Eq. 1.32) is different from the result of performing TD before TR (Eq. 1.33). TDk{TR[x[n]]} = TDk{x[-n]} (1.32) = x[-n+k] TR{TDk[x[n]]} = TR{x[n-k]} (1.33) = x[-n-k]

43. Figure 1.17 Graphical illustration of the different output signal generated by: (a) Time-delaying the input signal and then folding. (b) Folding the input signal and then time delaying.

44. 1.4.6 Time Scaling Let x[n] be a d-t signal, respectively. The signal y[n] obtained by scaling the independent variable, time n, by a factor of a is defined by y[n] = x[an] (1.34) Compressing a > 1 Stretching a < 1 This operation is known as “sampling rate alteration”. Let x[n] be a d-t signal with a sampling rate of Fx Hz. And y[n] be the d-t signal from altering the sampling rate Fx Hz. To a new sampling rate Fy Hz. The sampling rate alteration ration R can be expressed by

45. Fy = R Fx (1.35) If R is greater than 1, the process is called “interpolation” process and the operation is call “up-sampling”. Thus, the up-sampling operation will increase the number of samples in the input signal. If R is less than 1, on the other hand, the process is called “decimation” process and the operation is call “down-sampling”. The number of samples in the input signal will be decreased after the down-sampling process. To perform an up-sampling operation by an integer factor of L (L>1), L-1 new samples between successive values of the input signal will be interpolated.

46. This interpolation process can be accomplished in a various means. The easiest and very common mean to up-sampling the sequence is performed by adding L-1 zeros between successive values of the input signal x[n]. This up-sampling operation can be expressed by Eq. 1.36. (1.36) The decimation process by a factor of an integer M (M>1) can be performed by taking only every Mth. Sample of the input signal. This results in a signal with a lower sampling rate. The down-sampling operation by a factor of M is expressed by Eq. 1.37 xd[n] = x[Mn] (1.37)

47. (a) (b) Figure 1.18 Graphical illustration of the up-sampling operation. (a) the input signal. (b) Illustration of the output resulting from up-sampling the input in (a) by 4.

48. (a) (b) Figure 1.19 Graphical illustration of the down-sampling operation. (a) the input signal. (b) Illustration of the output resulting from down-sampling the input in (a) by 4.

49. Elementary Signals 1.5 There are some elementary signals that are used in studying signals and systems. Such signals are exponential and sinusoidal signals (See section 1.3), the impulse function, the step function, and ramp function. 1.5.1 Exponential Signals A exponential signal is a sequence of the form: x[n] = an (1.38) In the case that a are real, x[n] defined by Eq. 1.38 will be called “real exponential” signal. If a are complex numbers, a exponential signal x[n] defined by Eq. 1.38 will be called “complex exponential” signal.

50. Consider the case where “a” is expressed as: (1.40) Hence we can express x[n] as (1.41) That is, x[n] is complex values. The real part, xR[n], and imaginary part, xI[n], defined as following: (1.42) (1.43)