Slide - Sinais no Tempo Discreto - 2024-1

ELF51 - EL66D - PDS Prof. Daniel R. Pipa Sinais no Tempo Discreto Sinais no tempo discreto Sinais elementares Propriedades de sinais Processamento Digital de Sinais ELF51 - EL66D - Engenharia Eletrˆonica Sinais no Tempo Discreto Prof. Daniel R. Pipa danielpipa@utfpr.edu.br Representacao de sinais no tempo discreto ELFS1 - EL66D - PDS > Sinal no tempo discreto ou sequéncia: x[n] EC R, ne Z. Prof. Daniel R. Pipa SEER tenn erences) 1 n n= {@) > nZo 0, n<0 nj... —-2-1 0123 .... am]... 0 O15 4 yo. Mm={ 01345 +. } | x[n] 1012345 n ! The symbol + denotes the index n = 0; it is omitted when the table starts at n = 0. Impulso discreto e degrau unitario ELFS1 - EL66D - PDS aal® Del, Pipe Impulso discreto Degrau unitario 1, n=0 0, n<O — dn) = 2” uln|= 5 0, n#4#0 1, n>0 ln] Unit sample u[n] Unit step 1 1 0 rr (a) b) ELF51 - EL66D - PDS Prof. Daniel R. Pipa Sinais no Tempo Discreto Sinais no tempo discreto Sinais elementares Propriedades de sinais Sinoide Senoide x[n] = A cos(ω0n + φ), n ∈ Z, A, ω0, φ ∈ R ▶ A: amplitude ▶ ω0: frequˆencia ▶ φ: fase 26 Discrete-time signals and systems Unit step Unit sample 1 0 0 1 )b( )a( u[n] n n [n] δ ... Figure 2.2 Some elementary discrete-time signals. 0 0 10 20 n n 30 0 –1 0 1 10 20 30 –1 1 0 x[n] x[n] x[n] 0 1 5 10 0 < a <1 –1 < a < 0 15 (a) (b) n 20 25 30 Figure 2.3 Examples of a discrete-time sinusoidal signal (a), and two real exponential sequences (b). Sinusoidal sequence The real sinusoidal sequence has the general form x[n] = A cos(ω0n + φ), −∞ < n < ∞ (2.5) where A (amplitude) and φ (phase) are real constants. The quantity ω0 is the fre- quency of the sinusoid and has units of radians per sampling interval. The values of this sequence keep on oscillating between ±|A| as shown in Figure 2.3(a) for A = 1. Exponential sequence The exponential sequence has the general form defined by x[n] ≜ Aan, −∞ < n < ∞ (2.6) where A and a can take real or complex values. • If both A and a are real numbers in (2.6) then x[n] is termed as a real exponential sequence. For −1 < a < 1 (a > 1 or a < −1) the absolute value |x[n]| of the ELF51 - EL66D - PDS Prof. Daniel R. Pipa Sinais no Tempo Discreto Sinais no tempo discreto Sinais elementares Propriedades de sinais Exponenciais ▶ Exponencial real x[n] = Aan, n ∈ Z, A, a ∈ R 26 Discrete-time signals and systems Unit step Unit sample 1 0 0 1 )b( )a( u[n] n n [n] δ ... Figure 2.2 Some elementary discrete-time signals. 0 0 10 20 n n 30 0 –1 0 1 10 20 30 –1 1 0 x[n] x[n] x[n] 0 1 5 10 0 < a <1 –1 < a < 0 15 (a) (b) n 20 25 30 Figure 2.3 Examples of a discrete-time sinusoidal signal (a), and two real exponential sequences (b). Sinusoidal sequence The real sinusoidal sequence has the general form x[n] = A cos(ω0n + φ), −∞ < n < ∞ (2.5) where A (amplitude) and φ (phase) are real constants. The quantity ω0 is the fre- quency of the sinusoid and has units of radians per sampling interval. The values of this sequence keep on oscillating between ±|A| as shown in Figure 2.3(a) for A = 1. Exponential sequence The exponential sequence has the general form defined by x[n] ≜ Aan, −∞ < n < ∞ (2.6) where A and a can take real or complex values. • If both A and a are real numbers in (2.6) then x[n] is termed as a real exponential sequence. For −1 < a < 1 (a > 1 or a < −1) the absolute value |x[n]| of the ▶ Exponencial complexa x[n] = Ae jω0n = A cos(ω0n) + jA sin(ω0n) n ∈ Z, A, ω0 ∈ R ELF51 - EL66D - PDS Prof. Daniel R. Pipa Sinais no Tempo Discreto Sinais no tempo discreto Sinais elementares Propriedades de sinais Sequˆencia peri´odica Uma sequˆencia ´e peri´odica se x[n] = x[n + N], ∀n. Exemplo: x[n] = cos(ω0n + φ) x[n + N] = cos(ω0n + ω0N + φ), ser´a peri´odica se ω0N = 2πk, k ∈ Z. Ou seja, ω0 = 2π k N e a senoide ser´a peri´odica se sua frequˆencia for m´ultiplo racional de 2π. Soeue Mem Ne NaT:| imate Energia de uma sequéncia Poténcia de uma sequéncia Prof. Daniel R. Pipa oo 1 L Ey = x{nJ|? Py = lim | —— x(n] |? =D bbl = fim | SO bat STUY CRUSE N= OO n= aL Sequéncia de duracao finita Tem energia € finita, porém poténcia P zero. Sequéncia infinita Tem energia € infinita, porém poténcia P finita. Se a sequéncia for infinita e periddica, pode-se calcular sua poténcia em um periodo. ELF51 - EL66D - PDS Prof. Daniel R. Pipa Sinais no Tempo Discreto Sinais no tempo discreto Sinais elementares Propriedades de sinais Operac¸˜oes com sinais Sejam os sinais x1[n] e x2[n] ▶ Adic¸˜ao x1[n] + x2[n] ▶ Subtrac¸˜ao x1[n] − x2[n] ▶ Multiplicac¸˜ao x1[n] · x2[n] ▶ Divis˜ao x1[n]/x2[n] ▶ Escala a · x1[n] 30 Seja o sinal Discrete-time signals and systems 0 1 2 3 4 5 0 0 3 0 −2 −5 (a) (b) (c) (d) n n n n Folding Time-delay Time-advance Figure 2.4 Folding and time-shifting operations. Plotting To plot the sequence as a discrete-time signal (see Figure 2.3), we use the MATLAB function stem as follows stem(n,x,’fill’); ylabel(’x[n]’); xlabel(’n’); When the number of samples is large the resulting stem plot becomes unintelligible. In such cases, we plot the envelope of the discrete-time signal using the function plot with a statement like plot(n,x,’-’). This function “connects” the dots of the sequence with a straight line segment. This process, which is known as linear interpolation, is discussed in Chapter 12. Audio signals Although it is possible to plot audio (sound) signals, it is more informative to play and listen to these signals through a computer’s built-in audio input/output devices using appropriate MATLAB functions. The sound(x,Fs) plays the signal x as an audio through speakers at Fs Hz rate. To read a wave file from disk into signal x, the [x,Fs]=wavread(’wavefile’) can be used. Similarly, the wavwrite(x,Fs,’wavefle’) function is used to store x as a wave signal at Fs Hz rate. Additionally for Windows machines, the wavrecord and wavplay functions are avail- able to record and play, respectively, audio signals from a computer’s input/output devices. Tutorial Problem 6 discusses some of these functions. ▶ Revers˜ao temporal: x[−n] 30 Discrete-time signals and systems 0 1 2 3 4 5 0 0 3 0 −2 −5 (a) (b) (c) (d) n n n n Folding Time-delay Time-advance Figure 2.4 Folding and time-shifting operations. Plotting To plot the sequence as a discrete-time signal (see Figure 2.3), we use the MATLAB function stem as follows stem(n,x,’fill’); ylabel(’x[n]’); xlabel(’n’); When the number of samples is large the resulting stem plot becomes unintelligible. In such cases, we plot the envelope of the discrete-time signal using the function plot with a statement like plot(n,x,’-’). This function “connects” the dots of the sequence with a straight line segment. This process, which is known as linear interpolation, is discussed in Chapter 12. Audio signals Although it is possible to plot audio (sound) signals, it is more informative to play and listen to these signals through a computer’s built-in audio input/output devices using appropriate MATLAB functions. The sound(x,Fs) plays the signal x as an audio through speakers at Fs Hz rate. To read a wave file from disk into signal x, the [x,Fs]=wavread(’wavefile’) can be used. Similarly, the wavwrite(x,Fs,’wavefle’) function is used to store x as a wave signal at Fs Hz rate. Additionally for Windows machines, the wavrecord and wavplay functions are avail- able to record and play, respectively, audio signals from a computer’s input/output devices. Tutorial Problem 6 discusses some of these functions. ▶ Deslocamento: x[n − n0] 30 Discrete-time signals and systems 0 1 2 3 4 5 0 0 3 0 −2 −5 (a) (b) (c) (d) n n n n Folding Time-delay Time-advance Figure 2.4 Folding and time-shifting operations. Plotting To plot the sequence as a discrete-time signal (see Figure 2.3), we use the MATLAB function stem as follows stem(n,x,’fill’); ylabel(’x[n]’); xlabel(’n’); When the number of samples is large the resulting stem plot becomes unintelligible. In such cases, we plot the envelope of the discrete-time signal using the function plot with a statement like plot(n,x,’-’). This function “connects” the dots of the sequence with a straight line segment. This process, which is known as linear interpolation, is discussed in Chapter 12. Audio signals Although it is possible to plot audio (sound) signals, it is more informative to play and listen to these signals through a computer’s built-in audio input/output devices using appropriate MATLAB functions. The sound(x,Fs) plays the signal x as an audio through speakers at Fs Hz rate. To read a wave file from disk into signal x, the [x,Fs]=wavread(’wavefile’) can be used. Similarly, the wavwrite(x,Fs,’wavefle’) function is used to store x as a wave signal at Fs Hz rate. Additionally for Windows machines, the wavrecord and wavplay functions are avail- able to record and play, respectively, audio signals from a computer’s input/output devices. Tutorial Problem 6 discusses some of these functions.