This paper studies the performance of the generalized likelihood ratio test (GLRT) for the conditional signal model. By conditional signal model, we mean that under both hypotheses, the observations are a linear superposition of unknown deterministic signals corrupted by additive noise, with a mixing matrix depending on an unknown deterministic parameter vector. The contribution of this work is the derivation of closed form expressions for the probabilities of false alarm and of detection of the GLRT at high signal-to-noise ratio, allowing the receiver operating characteristic of the GLRT to be computed analytically. The most general case is tackled, i.e., when the number of unknown signals and the number of unknown deterministic parameters of the mixing matrix are allowed to be different under the two hypotheses.