Discrete-Time Signals and Systems

Discrete-Time Signals and Systems

• Discrete-time signals provide function values only at discrete time instances, forming a time-discontinuous sequence.
• In general, discrete-time signals have uniform intervals denoted by T, thus x(nT) represents the value at point nT, where n is an integer.

Discrete Time Signals

• Signals can be stored in memory for retrieval at any time and processed "non-real-time."
• It is acceptable to directly use x(n) to denote the sequence value at the nth discrete time point, representing the sequence as {x(n)}.
• To simplify, the sequence can be represented as x(n).
• Discrete-time signal (sequence) x(n) can be regarded as equidistant time sampling of analog signal x(t), specifically x(n) = x(t) | t=nT = x(nT).
• It's crucial that n is an integer for x(n) to be defined; x(n) is undefined when n is not an integer.
• Sampling analog signals helps understand this concept: x(n) = x(nT) means sampling x(t) when t = nT.
• Not sampling at moments between adjacent samples is not equal to zero.
• Meeting sampling theorem requirements enables signal recovery between sampling points with interpolation via low-pass smoothing filters.
• For a sequence x(n), y(n) = x(n/m) only has definition for integer values of n/m, where n/m must be an integer.
• Analog signals can also be sampled at non-equidistant time intervals, this is excluded in this text.
• Sequences are expressed in function, number sequence, and graphical forms.
• Function representation: x(n) = a^n * u(n).
• Numerical representation: x(n) = {…,-5, -3, -1, 0, 2, 7, 9,…}, where underlined value indicates x(0), e.g., x(-1)=-3, x(0)=-1, x(1)=0, and so on.
• Graphical representation as shown in Figure 1.1 is also used.

Sequence Operations

• Signal processing achieves enhanced signal processing through computation.
• Sequence operations involve three basic units: adders, multipliers, and delay units.
• Three categories of sequence operations exist: amplitude, variable 'n', and combined amplitude and variable 'n' operations.

Amplitude-Based Operations

• Addition combines two sequences by adding the sequence values of the same index (n) to form a new sequence: z(n) = x(n) + y(n).
• Multiplication combines two sequences by multiplying the sequence values of the same index (n) to form a new sequence: w(n) = x(n) * y(n).
• Scalar multiplication involves multiplying x(n) or y(n) by a constant c, expressed as w(n) = c * x(n).
• Accumulation represents the sum of x(n) values up to a certain n, given by y(n) = the sum of x(k) from k = -∞ to n.
• Absolute summability is defined as S = the sum of |x(n)| from n = -∞ to ∞.
• When S = B < ∞, sequence x(n) is absolutely summable, which ensures existence of the Fourier transform and system stability.
• Sequence energy is defined as E[x(n)] = the sum of |x(n)|^2 from n = -∞ to ∞.
• If E[x(n)] = A < ∞, x(n) is an energy-limited signal and finite-length sequences and absolutely summable infinite-length sequences are energy signals.
• Average power is defined as P[x(n)] = the limit as N tends to infinity of (1/(2N+1)) * the sum of |x(n)|^2 from n = -N to N.
• If this limit exists, and P[x(n)] = C < ∞, x(n) is a power-limited signal.
• Periodic and random signals have infinite duration, making them power signals rather than energy signals.
• For periodic signals, average power is calculated over one single period N using the equation P[x(n)] = (1/N) * the sum of |x(n)|^2 from n=0 to N-1.

Operations Based on Variable n

• Shifting a sequence x(n) results in x(n-m).
• For m > 0, the sequence shifts to the right (delay).
• For m < 0, the sequence shifts to the left (advance) by |m|.
• Folding a sequence x(n) yields x(-n), which is the reflection around the vertical axis at n = 0.
• The folded sequence x(n-m) about n=0 is x[-(n+m)] = x(-n-m).
• To obtain x(-n-m) from x(n), first fold to get x(-n), then shift x(-n) left by m if m is positive, or right by |m| if m is negative.
• Time-scale transformation changes the sampling rate of analog signals, and is covered in detail in Chapter 9.
• Decimation (down-sampling) reduces the sampling frequency, defined as xa(n) = x(Dn), with D being an integer.
• Interpolation (up-sampling) increases the sampling frequency. Inserting zero values (a step in interpolation) is shown in the equation.
• A system represented by decimation and interpolation operations is a linear shift-varying system because it involves compression or expansion on the time axis.
• Decimation and interpolation are fundamental to multi-rate digital signal processing.

Operations Involving Both Amplitude and Variable n

• Differencing involves both amplitude and variable operations.
• Forward difference: Δx(n) = x(n+1) - x(n)
• Backward difference: ∇x(n) = x(n) - x(n-1)
• From above: ∇x(n) = x(n) - x(n-1)
• Convolution Sum: y(n) = x(n) * h(n) = the sum of x(m)h(n-m) , and the sum of x(n-m)h(m) from m = -∞ to ∞.
• Discussed in Section 1.1.3.
• Correlation operations.
• Conjugate symmetric components from the complex sequence x(n):
  • x*(n) represents the conjugate symmetric part of x(n).
  • xe(n) = 1/2[x(n) + x*(-n)] applies to complex sequences x(n).
  • xo(n) = 1/2[x(n) - x*(-n)]
• Real sequences are split into even and odd symmetric components:
  • xa(n) represents 对实序列 x(n).
  • xe(n) = 1/2[x(n) + x(-n)] applies to real sequences x(n).
  • xo(n) = 1/2[x(n) - x(-n)]
• Also: x(n) = xe(n) + xo(n),
• Convolution is crucial for determining the output response of linear time-invariant systems when subjected to an input signal.

Convolution Sum and its Steps:

1. Folding: Select m as the dummy variable and plot x(m) and h(m). Flip h(m) about vertical axis at m=0 to get h(-m).
2. Shifting: Shift h(-m) by n to obtain h(n-m). Shift to the right if n>0, left by |n| if n<0.
3. Multiplication: Multiply h(n-m) and x(m) at each corresponding m.
4. Summation: Sum all products from step above to get one y(n) value, repeat for all n values to comprise entire y(n) list.

• Three common methods exist for computing convolution sums: graphical, analytical, and combined approaches, dividing the problem into intervals if required.
• In discrete-time linear time-invariant systems, convolution provides a vital operation for finding the zero-state response, demanding skill in mastering.

Properties

• Commutative: x(n) * h(n) = h(n) * x(n)
• Associative: [x(n) * h1(n)] * h2(n) = x(n) * [h1(n) * h2(n)]
• Distributive: x(n) * [h1(n) + h2(n)] = x(n) * h1(n) + x(n) * h2(n)

Sequence Correlation

• Correlation in statistical communication and digital signal processing is a vital concept that correlates with signal power spectrum.
• Correlation functions are used to analyze the power spectral density of stationary random signals, offering insights into determinant signals.
• Correlation measures the relationship between two deterministic or random signals.
• The correlation functions of balanced random signals can be determined through statistical analysis, describing statistical characteristics in digital random signal processing.
• Discussion focuses on the foundational concepts of deterministic signal correlation, as the study of correlation in greater depth pertains to "Random Signal Processing."

Cross-Correlation Function Sequence

• This is defined for real signals x(n) and y(n) as rxy(m) = the sum of x(n) * y(n-m) from n = -∞ to ∞.
  • For m>0, shift y(n) right by m.
  • For m<0, shift y(n) left by |m|.

Key Differences

• Convolution involves folding, shifting, multiplying, and summing, whereas correlation omits folding of the sequences.
• Correlation lacks the commutative property.
• This suggests interval m in correlation functions indicates x sequence's variable being subtracted by y sequence's variable.

Traits of Cross-Correlation Functions

• 𝑟xy(m) and 𝑟yx(m) exhibit even symmetry with regard to m=0, as derived from the expressions.
  • 𝑟xy(m) = 𝑟yx(-m)
• 𝑟xy(m) is not an even function 𝑟xy(m) ≠ 𝑟yx(-m)
• For absolutely summable energy signals x(n), y(n), the limit of 𝑟xy(m) =0 as |m| tends to infinity.

Variable m's Interval Range in 𝑟xy(m)

• In finite-length sequences x(n), y(n), define them as:
  • x(n) defined for: N1 ≤ n ≤ N2
  • y(n) defined for: N3 ≤ n ≤ N4
• The value of 𝑟xy(m) relies on left or right shifting of y(n), and is summed with x(n):
  • When m>0, y(n) shifts right by m, maximizing y(n) with 𝑟xy(m):
    • N2 - Ny1
  • When m0 results from shifting y(n) to the left.
  • if N2 N3 then 𝑟xy(m) only exists for m ≥ 0.
  • if Ny1 ≥ N2 then 𝑟xy(m) only exists for m≤0
  • when N1 = N3 =0, x(n), y(n) are the causal finite-length sequences starting from n=0 length (x) and length (y) indicate causal finite sequences x(n) , y(n) then the 𝑟xy(m) is given as follows: From -(length(y) -1) to less than or equal to the length(x) -1

Correlation Operation

• Can be expressed as convolution using the equation: This means convolving x(m) with y(-m).

Autocorrelation Function Sequence

• In autocorrelation, x(n) = y(n):
• Denoted as rxx(m) = the sum of x(n) * x(n-m) from n = -∞ to ∞. Equal to x(m) * x(-m).

Characteristics

1. 𝑟xx(m) exhibits even symmetry, implying it's a real and even sequence: rxx(m)=rxx(-m).
2. When m=0, the autocorrelation sequence attains its maximum which is a sign of self similarity, : 𝑟xx(0) = the sum of x^2(n) which is greater than |rxx(m)| . Value at 0 and signal energy equates.
3. When x(n) denotes absolute energy signal, the lim m tends to infinity resulting in : 𝑟xx(m) or m is = 0

Characteristics of Correlation Functions

• Correlation includes autocorrelation and cross-correlation and it involves convolution to solve it, but convolution requires a flipping before shifting. It is the correlation's representation involving convolution, x(-m)- it takes flips before during convolution. Doing it, the effects result in non-flipping in the correlation.

Significance

• Functions indicate relations between signals with convolution indicating signal transformation through systems different from physical attributes.
• Mean is used when expressing the mutual correlation and autocorrelation for signals with power.

Aperiodic Sequences

• Where x and y are periodic, the mean is used over a given period to represent the mutual correlation:r
• xx (m)= 1/N ( the sums of n to -0, 1 (x(n) (x(n-m)

Typical Sequences

1. Use sampling (unit samples, and unit impulse) Sequence

• if and only if the sequence (n=0) is (1, otherwise, it is not equal (N 0

2. Unit test Sequence u(n)->when = more or is more than to go on, it is one (greater than N or equal to a otherwise, it is less than 0.

• NOTE: the source of the material that you will provide in the information that follows will be the document you provided. You should do your best to summarize and shorten what is present in it*