Q3Digital Signal Processing

(a) Linear Convolution Calculation

The given signal x(n) = sum from k=0 to 2 of delta(n-k) = delta(n) + delta(n-1) + delta(n-2) represents the rectangular pulse sequence x(n) = {1, 1, 1} for n=0, 1, 2, and x(n)=0 elsewhere. Since only this single sequence formula is explicitly given for what the problem calls 'two discrete time signals,' the most direct and standard interpretation is to compute the self-convolution of this sequence with itself, y(n) = x(n) * x(n), which is a common textbook exercise illustrating how convolving a rectangular pulse with itself produces a triangular result.

The linear convolution sum is defined as:

Since x(k) is non-zero only for k=0,1,2, the sum reduces to k ranging over these three values for each n, with each term contributing only when both x(k) and x(n-k) are non-zero (i.e., both k and n-k lie in {0,1,2}).

• n=0: only k=0 contributes (n-k=0 is valid): y(0) = x(0)x(0) = 11 = 1
• n=1: k=0 (n-k=1) and k=1 (n-k=0) contribute: y(1) = x(0)x(1) + x(1)x(0) = 1+1 = 2
• n=2: k=0 (n-k=2), k=1 (n-k=1), k=2 (n-k=0) contribute: y(2) = x(0)x(2)+x(1)x(1)+x(2)*x(0) = 1+1+1 = 3
• n=3: k=1 (n-k=2) and k=2 (n-k=1) contribute: y(3) = x(1)x(2)+x(2)x(1) = 1+1 = 2
• n=4: only k=2 contributes (n-k=2): y(4) = x(2)*x(2) = 1

Result: y(n) = {1, 2, 3, 2, 1} for n=0,1,2,3,4, and zero elsewhere - a symmetric triangular sequence. This result directly illustrates the general principle that convolving a rectangular pulse of length N with itself produces a triangular pulse of length 2N-1 (here N=3, giving length 2*3-1=5, matching the computed result), a standard, easily-verified relationship in discrete-time signal processing.

Convolution as the Fundamental Input-Output Relationship of LTI Systems

The convolution operation demonstrated above is not merely an abstract mathematical exercise - it is the single most fundamental result in the theory of linear time-invariant (LTI) discrete-time systems. Any LTI system is completely and uniquely characterized by its impulse response h(n), defined as the system's output when the input is the unit impulse delta(n). Because any arbitrary input sequence x(n) can itself be expressed as a weighted sum of time-shifted impulses, x(n) = sum over k of x(k)delta(n-k) (the sifting/sampling property of the discrete impulse), linearity and time-invariance together imply that the system's response to this arbitrary input must be the identical weighted sum of time-shifted copies of the impulse response, i.e., y(n) = sum over k of x(k)h(n-k) = x(n)*h(n). This is precisely why the convolution sum is called the fundamental input-output relationship for LTI systems: once h(n) is known (from the system's physical description or difference equation), the output for ANY input signal can be computed purely by convolution, without needing to re-derive or re-solve the underlying difference equation for each new input.

Graphical/Tabular Convolution Method

Besides the direct algebraic summation approach used in part (a), convolution is frequently performed using the graphical (fold-shift-multiply-sum) method, which is especially useful for building intuition and for hand-sketching convolution results: one sequence (say h(k), viewed as a function of the dummy index k) is first time-reversed (folded) to obtain h(-k), then shifted right by n samples to obtain h(n-k) for the particular output index n being computed; this shifted, folded sequence is then multiplied, sample-by-sample, against the other sequence x(k), and all the resulting products are summed to give the single output value y(n); the entire process is then repeated with a different shift amount n to obtain each subsequent output sample. Equivalently, the tabular (array multiplication) method arranges the two sequences along the rows and columns of a table, forming a grid of all pairwise products x(k)*h(m), after which the final convolution output at each index n is obtained by summing all table entries lying along the corresponding anti-diagonal (where k+m=n) - a mechanical, easily-verified bookkeeping method well suited to problems, such as part (a) above, involving short, finite-length sequences.

(b) Properties of Linear Convolution

• Commutativity: x(n)h(n) = h(n)x(n) - the order of the two sequences being convolved does not affect the result, since the convolution sum can be re-indexed (substituting m=n-k) to show both orderings yield an identical expression.
• Associativity: (x(n)h1(n))h2(n) = x(n)(h1(n)h2(n)) - convolving with a cascade of two systems in sequence is equivalent to convolving with a single system whose impulse response is itself the convolution of the two individual impulse responses; this property directly justifies the cascade-system-combination technique used throughout LTI system analysis.
• Distributivity over addition: x(n)(h1(n)+h2(n)) = x(n)h1(n) + x(n)*h2(n) - convolving a signal with the sum of two impulse responses (equivalent to two systems connected in parallel, with outputs summed) equals the sum of convolving the signal separately with each impulse response.
• Identity element (unit impulse): x(n)*delta(n) = x(n) - convolving any sequence with the unit impulse function leaves the sequence unchanged, since the unit impulse acts as the identity element under the convolution operation, directly following from the sifting property of the delta function.
• Length property: convolving a sequence of length N1 with a sequence of length N2 (both finite-length) produces a result of length N1+N2-1, a direct consequence of the convolution sum's range of non-zero output indices, exactly as demonstrated numerically in part (a) above (N1=N2=3, giving output length 3+3-1=5).
• Finite-support property: if both sequences being convolved have finite support (i.e., are non-zero only over a finite range of indices), the convolution result is also guaranteed to have finite support, with its non-zero region's start and end indices given by the sum of the two individual sequences' respective start and end indices; this property is what permits convolution of finite-length sequences to be computed exactly using a finite number of arithmetic operations, exactly as done in part (a).
• Convolution and stability: an LTI system is bounded-input bounded-output (BIBO) stable if and only if its impulse response h(n) is absolutely summable, i.e., sum over all n of |h(n)| is finite - this criterion follows directly from the convolution sum, since a bounded input convolved with an absolutely summable impulse response can never produce an unbounded output, whereas a non-absolutely-summable impulse response can be shown (by an appropriately chosen bounded input, typically a sign sequence matching h(n)'s sign pattern) to produce an unbounded output.
• Convolution and causality: an LTI system is causal if and only if its impulse response h(n) is zero for all n<0 (i.e., h(n)=0 for negative indices) - this ensures that, in the convolution sum y(n)=sum over k of x(k)*h(n-k), the output at any time n depends only on input values x(k) for k<=n (present and past inputs), never on future input values, directly reflecting the physical requirement that a causal system's output cannot depend on inputs that have not yet occurred.